1579 lines
64 KiB
TeX
1579 lines
64 KiB
TeX
\documentclass{article} \usepackage{graphicx}
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\usepackage[francais]{babel} \usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amsfonts}
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\usepackage{verbatim} \usepackage{float} \usepackage{hyperref}
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\usepackage{scrtime} \usepackage{lifecon} \usepackage{icomma} \usepackage{fullpage} \usepackage[scale=2]{ccicons}
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\usepackage{tabularx}
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\begin{document}
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\title{Corrigé des exercices supplémentaires \\ Décroissances multiples et statuts conjoints \\ ACT-2007 \\ Mathématiques actuarielles vie II}
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\author{François Pelletier}
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\date{\today}
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\maketitle
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\tableofcontents
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\clearpage
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\section{Exercice 1}
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\label{sec:ex1}
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\subsection{Partie a}
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\label{sec:ex1pta}
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\subsubsection{Approche DUD}
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On utilise les formules pour $m=3$ de la section 9.6.3
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\begin{align*}
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q_{x}^{(1)} &= q_{x}^{'(1)} \left(1-\frac{1}{2} (q_{x}^{'(2)}+q_{x}^{'(3)}) +\frac{1}{3}(q_{x}^{'(2)} \cdot q_{x}^{'(3)}) \right) \\
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q_{x}^{(2)} &= q_{x}^{'(2)} \left(1-\frac{1}{2} (q_{x}^{'(1)}+q_{x}^{'(3)}) +\frac{1}{3}(q_{x}^{'(1)} \cdot q_{x}^{'(3)}) \right) \\
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q_{x}^{(3)} &= q_{x}^{'(3)} \left(1-\frac{1}{2} (q_{x}^{'(1)}+q_{x}^{'(2)}) +\frac{1}{3}(q_{x}^{'(1)} \cdot q_{x}^{'(2)}) \right)
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\end{align*}
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On peut alors calculer la table à plusieurs décroissances
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\begin{center}\begin{tabular}{|l|l|l|l|}
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\hline
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\multicolumn{1}{|l|}{x} & \multicolumn{1}{l|}{$q_x^{(1)}$ DUD} & \multicolumn{1}{l|}{$q_x^{(2)}$ DUD} & \multicolumn{1}{l|}{$q_x^{(3)}$ DUD} \\ \hline
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25 & 0,004994502 & 0,000996902 & 0,001196402 \\ \hline
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26 & 0,0049940023 & 0,0009968023 & 0,0013958023 \\ \hline
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27 & 0,004991255 & 0,001993505 & 0,001494755 \\ \hline
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28 & 0,004988508 & 0,002990108 & 0,001593608 \\ \hline
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29 & 0,004983015 & 0,004983015 & 0,001791015 \\ \hline
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30 & 0,0049827658 & 0,0049827658 & 0,0018905158 \\ \hline
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31 & 0,004980269 & 0,005979319 & 0,001889569 \\ \hline
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32 & 0,00498002 & 0,00597902 & 0,00198902 \\ \hline
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33 & 0,004974778 & 0,007971628 & 0,002086378 \\ \hline
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34 & 0,0049717845 & 0,0089671845 & 0,0022839345 \\ \hline
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\end{tabular}\end{center}
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\subsubsection{Approche force constante}
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On utilise la formule (voir section 9.6.2 p.211)
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\begin{align*}
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q_x^{(j)} = \frac{-\ln(p_x^{'(j)})}{\sum_{j=1}^{m} -\ln(p_x^{'(j)})} \underbrace{\left(1-\prod_{j=1}^m p_x^{'(j)} \right)}_{q_x^{(\tau)}}
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\end{align*}
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On sait que $p_x^{'(j)} = 1 - q_x^{'(j)}$.
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On évalue d'abord $\sum_j^m(-\ln(p_x^{'(j)})$
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\begin{center}
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\begin{tabular}{|l|l|l|l|l|}
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\hline
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\multicolumn{1}{|l|}{x} & \multicolumn{1}{l|}{$-\ln(p_x^{'(1)})$} & \multicolumn{1}{l|}{$-\ln(p_x^{'(2)})$} & \multicolumn{1}{l|}{$-\ln(p_x^{'(3)})$} & \multicolumn{1}{l|}{$\sum_j^m(-\ln(p_x^{'(j)})$} \\ \hline
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25 & 0,0050125418 & 0,0010005003 & 0,0012007206 & 0,0072137627 \\ \hline
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26 & 0,0050125418 & 0,0010005003 & 0,0014009809 & 0,0074140231 \\ \hline
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27 & 0,0050125418 & 0,0020020027 & 0,0015011261 & 0,0085156706 \\ \hline
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28 & 0,0050125418 & 0,003004509 & 0,0016012814 & 0,0096183322 \\ \hline
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29 & 0,0050125418 & 0,0050125418 & 0,0018016219 & 0,0118267056 \\ \hline
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30 & 0,0050125418 & 0,0050125418 & 0,0019018073 & 0,0119268909 \\ \hline
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31 & 0,0050125418 & 0,0060180723 & 0,0019018073 & 0,0129324214 \\ \hline
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32 & 0,0050125418 & 0,0060180723 & 0,0020020027 & 0,0130326168 \\ \hline
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33 & 0,0050125418 & 0,0080321717 & 0,0021022081 & 0,0151469216 \\ \hline
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34 & 0,0050125418 & 0,0090407447 & 0,0023026491 & 0,0163559355 \\ \hline
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\end{tabular}\end{center}
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On évalue ensuite $q_x^{(tau)}$
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\begin{center}
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\begin{tabular}{|l|l|l|}
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\hline
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\multicolumn{1}{|l|}{x} & \multicolumn{1}{l|}{$p_x^{(tau)}$} & \multicolumn{1}{l|}{$q_x^{(tau)}$} \\ \hline
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25 & 0,992812194 & 0,007187806 \\ \hline
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26 & 0,992613393 & 0,007386607 \\ \hline
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27 & 0,991520485 & 0,008479515 \\ \hline
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28 & 0,990427776 & 0,009572224 \\ \hline
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29 & 0,988242955 & 0,011757045 \\ \hline
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30 & 0,9881439525 & 0,0118560475 \\ \hline
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31 & 0,987150843 & 0,012849157 \\ \hline
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32 & 0,98705194 & 0,01294806 \\ \hline
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33 & 0,984967216 & 0,015032784 \\ \hline
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34 & 0,9837770965 & 0,0162229035 \\ \hline
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\end{tabular}\end{center}
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On peut maintenant calculer la table à plusieurs décroissances
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\begin{center}
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\begin{tabular}{|l|l|l|l|}
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\hline
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\multicolumn{1}{|l|}{x} & \multicolumn{1}{l|}{$q_x^{(1)}$ FC} & \multicolumn{1}{l|}{$q_x^{(2)}$ FC} & \multicolumn{1}{l|}{$q_x^{(3)}$ FC} \\ \hline
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25 & 0,0049945056 & 0,0009969003 & 0,0011964001 \\ \hline
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26 & 0,0049940061 & 0,0009968006 & 0,0013958003 \\ \hline
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27 & 0,0049912597 & 0,0019935026 & 0,0014947527 \\ \hline
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28 & 0,0049885128 & 0,0029901061 & 0,0015936052 \\ \hline
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29 & 0,0049830174 & 0,0049830174 & 0,0017910102 \\ \hline
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30 & 0,0049827683 & 0,0049827683 & 0,0018905109 \\ \hline
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31 & 0,0049802689 & 0,0059793254 & 0,0018895626 \\ \hline
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32 & 0,00498002 & 0,0059790265 & 0,0019890135 \\ \hline
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33 & 0,0049747705 & 0,0079716463 & 0,0020863672 \\ \hline
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34 & 0,004971772 & 0,0089672112 & 0,0022839203 \\ \hline
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\end{tabular}\end{center}
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\subsection{Partie b}
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\label{sec:ex1ptb}
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On définit le montant de la prestation
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\begin{eqnarray*}
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B^{(1)} &=& 200000 \qquad 0 \leq k < 10 \\
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B^{(2)} &=& \begin{cases}
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500000 \qquad 0 \leq k < 5 \\
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100000 \qquad 5 \leq k < 10
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\end{cases} \\
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B^{(3)} &=& 100000 \qquad 0 \leq k < 10
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\end{eqnarray*}
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\textbf{--- Coût du produit ---}
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\begin{align*}
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A^{TOT} &= 200000 \cdot \sum_{t=0}^{9} {}_{t}p_{x}^{(\tau)}q_{x+t}^{(1)}v^{t+1} \\
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&+ 500000 \cdot \sum_{t=0}^{4} {}_{t}p_{x}^{(\tau)}q_{x+t}^{(2)}v^{t+1} \\
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&+ 100000 \cdot \sum_{t=5}^{9} {}_{t}p_{x}^{(\tau)}q_{x+t}^{(2)}v^{t+1} \\
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&+ 100000 \cdot \sum_{t=0}^{9} {}_{t}p_{x}^{(\tau)}q_{x+t}^{(3)}v^{t+1}
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\end{align*}
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On a donc
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\begin{center}\begin{tabular}{|l|l|}
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\hline
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\multicolumn{1}{|l|}{t} & \multicolumn{1}{l|}{${}_{t}p_{x}^{\tau}$} \\ \hline
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0 & 1 \\ \hline
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1 & 0,992812194 \\ \hline
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2 & 0,9854786805 \\ \hline
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3 & 0,9771222992 \\ \hline
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4 & 0,9677690657 \\ \hline
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5 & 0,9563909613 \\ \hline
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6 & 0,9450519446 \\ \hline
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7 & 0,9329088238 \\ \hline
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8 & 0,9208294644 \\ \hline
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9 & 0,9069868339 \\ \hline
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10 & 0,892272874 \\ \hline
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\end{tabular}\end{center}
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Calcul des éléments de la somme
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\begin{center}\begin{tabular}{|l|l|l|}
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\hline
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\multicolumn{1}{|l|}{$B^{(1)} v^{(t+1)} {}_{t}p_{x}^{(\tau)} q_{x+t}^{(1)}$ DUD} & \multicolumn{1}{l|}{$B^{(2)} v^{(t+1)} {}_{t}p_{x}^{(\tau)} q_{x+t}^{(2)}$ DUD} & \multicolumn{1}{l|}{$B^{(3)} v^{(t+1)} {}_{t}p_{x}^{(\tau)} q_{x+t}^{(3)}$ DUD} \\ \hline
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969,8062135922 & 483,9330097087 & 116,1555339806 \\ \hline
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934,6981644637 & 466,4141349519 & 130,6220734232 \\ \hline
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900,2752547397 & 898,9238286262 & 134,8048675532 \\ \hline
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866,1651269989 & 1297,9468387946 & 138,3507529412 \\ \hline
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831,9702610334 & 2079,9256525835 & 149,5150242438 \\ \hline
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798,2015894875 & 399,1007947437 & 151,4232048617 \\ \hline
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765,3813997351 & 459,459071958 & 145,1970733023 \\ \hline
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733,5039986998 & 440,3230386932 & 146,4807494241 \\ \hline
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702,179078763 & 562,5889642997 & 147,2438551018 \\ \hline
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671,0749495924 & 605,1803820463 & 154,1389444233 \\ \hline
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\end{tabular}\end{center}
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Le coût est de $17280,9838327667$ sous DUD
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\begin{center}\begin{tabular}{|l|l|l|}
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\hline
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\multicolumn{1}{|l|}{$B^{(1)} v^{(t+1)} {}_{t}p_{x}^{(\tau)} q_{x+t}^{(1)}$ FC} & \multicolumn{1}{l|}{$B^{(2)} v^{(t+1)} {}_{t}p_{x}^{(\tau)} q_{x+t}^{(2)}$ FC} & \multicolumn{1}{l|}{$B^{(3)} v^{(t+1)} {}_{t}p_{x}^{(\tau)} q_{x+t}^{(3)}$ FC} \\ \hline
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969,8069078907 & 483,9321890038 & 116,1553509723 \\ \hline
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934,6988712335 & 466,4133314942 & 130,6218807299 \\ \hline
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900,2761023619 & 898,9227545224 & 134,8046585628 \\ \hline
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866,1659567508 & 1297,9459931435 & 138,3505071955 \\ \hline
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831,9706621021 & 2079,9266552552 & 149,5146231752 \\ \hline
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798,2019830011 & 399,1009915006 & 151,4228113481 \\ \hline
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765,3813911038 & 459,4595646731 & 145,1965849028 \\ \hline
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733,5039972184 & 440,3235186603 & 146,4802701977 \\ \hline
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702,1780175427 & 562,5902583452 & 147,2430916664 \\ \hline
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671,0732619681 & 605,182182804 & 154,1379874777 \\ \hline
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\end{tabular}\end{center}
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Le coût est de $17280,9823568037$ sous FC
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\textbf{--- Coût de la rente associée au paiement des primes ---}
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\begin{align*}
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\ddot{a}^{Primes} &= \sum_{t=0}^4 {}_{t}p_{x}^{\tau}v^t \\
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&= 4,6468591093
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\end{align*}
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On calcule la prime selon la formule
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\begin{align*}
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\pi = \frac{A^{TOT}}{\ddot{a}^{Primes}}
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\end{align*}
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La prime est donc 3718,8525466621 sous DUD et 3718,8522290361 sous FC.
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\textbf{ --- Calcul de la réserve ---}
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On a l'équation de la perte prospective au temps $j$
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\begin{align*}
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{}_{j}L &= \underbrace{b_{k+1}v^{k+1-j}}_{\text{Partie prestation: } Z_j} - \underbrace{\sum_{i=j}^{k} \pi_i v^{i-j}}_{\text{Partie primes: } Y_j} \qquad k=j, j+1, \ldots \\
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{}_{2}V &= E[{}_{2}L | K \geq 2] \\
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&= E[Z_2 | K \geq 2] - E[Y_2 | K \geq 2] \\
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\end{align*}
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\textbf{Partie prestation}
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Pour $t=2$
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\begin{align*}
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E[Z_2 | K \geq 2] &= \frac{1}{{}_{2}p_x^{(\tau)}v^{2}} \bigg[200000 \cdot \sum_{t=2}^{9} {}_{t}p_{x}^{(\tau)}q_{x+t}^{(1)}v^{t+1} \\
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&+ 500000 \cdot \sum_{t=2}^{4} {}_{t}p_{x}^{(\tau)}q_{x+t}^{(2)}v^{t+1} \\
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&+ 100000 \cdot \sum_{t=5}^{9} {}_{t}p_{x}^{(\tau)}q_{x+t}^{(2)}v^{t+1} \\
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&+ 100000 \cdot \sum_{t=2}^{9} {}_{t}p_{x}^{(\tau)}q_{x+t}^{(3)}v^{t+1} \bigg]
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\end{align*}
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Pour $t=7$
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\begin{align*}
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E[Z_7 | K \geq 7] &= \frac{1}{{}_{7}p_x^{(\tau)}v^{7}} \bigg[200000 \cdot \sum_{t=7}^{9} {}_{t}p_{x}^{(\tau)}q_{x+t}^{(1)}v^{t+1} \\
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&+ 100000 \cdot \sum_{t=7}^{9} {}_{t}p_{x}^{(\tau)}q_{x+t}^{(2)}v^{t+1} \\
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&+ 100000 \cdot \sum_{t=7}^{9} {}_{t}p_{x}^{(\tau)}q_{x+t}^{(3)}v^{t+1} \bigg]
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\end{align*}
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\textbf{Partie primes}
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Pour $t=2$
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\begin{align*}
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E[Y_2 | K \geq 2] = \pi \left(\sum_{t=2}^4 {}_{t}p_{x}^{\tau}v^t\right) \cdot \frac{1}{{}_{2}p_x^{(\tau)}v^{2}}
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\end{align*}
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Pour $t=7$ (les primes sont payées en entier)
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\begin{align*}
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E[Y_7 | K \geq 7] = 0
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\end{align*}
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On obtient les résultats
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\begin{center}\begin{tabular}{|l|l|l|l|l|}
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\hline
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{} & Réserve à $t=2$ & {} & Réserve à $t=7$ & {} \\ \hline
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{} & DUD & FC & DUD & FC \\ \hline
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Partie prestation & 15264,5386467762 & 15264,5377024773 & 5487,7957838611 & 5487,7939709539 \\ \hline
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Partie primes & 10741,1550043718 & 10741,1540869733 & 0 & 0 \\ \hline
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Réserve & 4523,3836424044 & 4523,383615504 & 5487,7957838611 & 5487,7939709539 \\ \hline
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\end{tabular}\end{center}
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\textbf{--- Variance de la perte à l'émission ---}
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On définit la perte à l'émission et la variance de la perte à l'émission
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\begin{align*}
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L &= Z - \pi Y \\
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V[L] &= E[L^2] - E[L]^2 \\
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&= E[L^2] \\
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&= E[(Z - \pi Y)(Z - \pi Y)] \\
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&= E[Z^2] + \pi^2 E[Y^2] - 2 \pi E[YZ]
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\end{align*}
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On définit $Z$ et $Y$
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\begin{align*}
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Z &= v^{k+1} \begin{cases} 200000 & (1) \\ 500000 & (2), k < 5 \\ 100000 & (2), k \geq 5 \\ 100000 & (3) \end{cases} \\
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Y &= \pi \begin{cases} \ddot{a}_{\lcroof{k+1}} & k < 5 \\ \ddot{a}_{\lcroof{5}} & k \geq 5 \end{cases} \\
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\end{align*}
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On développe l'expression des 3 premières espérances
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\begin{align*}
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E[Z^2] &= 200000^2 \cdot \sum_{k=0}^{9} {}_{k}p_{x}^{(\tau)}q_{x+k}^{(1)}v^{2(k+1)} \\
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&+ 500000^2 \cdot \sum_{k=0}^{4} {}_{k}p_{x}^{(\tau)}q_{x+k}^{(2)}v^{2(k+1)} \\
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&+ 100000^2 \cdot \sum_{k=5}^{9} {}_{k}p_{x}^{(\tau)}q_{x+k}^{(2)}v^{2(k+1)} \\
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&+ 100000^2 \cdot \sum_{k=0}^{9} {}_{k}p_{x}^{(\tau)}q_{x+k}^{(3)}v^{2(k+1)} \\
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\end{align*}
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\begin{align*}
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E[Y^2] &= \pi^2 \left(\sum_{k=0}^4 (\ddot{a}_{\lcroof{k}})^2 {}_{k}p_{x}^{\tau}q_{x+k}^{(\tau)} + (\ddot{a}_{\lcroof{5}})^2 {}_{5}p_{x}^{\tau} \right) \\
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&= \pi^2 \left(\sum_{k=0}^4 (\frac{1-v^{k}}{d})^2 {}_{k}p_{x}^{\tau}q_{x+k}^{(\tau)} + (\ddot{a}_{\lcroof{5}})^2 {}_{5}p_{x}^{\tau} \right) \\
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\end{align*}
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\begin{align*}
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E[ZY] &= 200000\pi \sum_{k=0}^4 {}_{k}p_{x}^{\tau} q_{x+k}^{(1)} v^{k+1} \ddot{a}_{\lcroof{k+1}} \\
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&+ 200000\pi\ddot{a}_{\lcroof{5}}\sum_{k=5}^{9} {}_{k}p_{x}^{(\tau)}q_{x+k}^{(1)}v^{k+1}\\
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&+ 500000\pi \sum_{k=0}^4 {}_{k}p_{x}^{\tau} q_{x+k}^{(2)} v^{k+1} \ddot{a}_{\lcroof{k+1}} \\
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&+ 100000\pi\ddot{a}_{\lcroof{5}}\sum_{k=5}^{9} {}_{k}p_{x}^{(\tau)}q_{x+k}^{(2)}v^{k+1}\\
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&+ 100000\pi \sum_{k=0}^4 {}_{k}p_{x}^{\tau} q_{x+k}^{(3)} v^{k+1} \ddot{a}_{\lcroof{k+1}} \\
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&+ 100000\pi\ddot{a}_{\lcroof{5}}\sum_{k=5}^{9} {}_{k}p_{x}^{(\tau)}q_{x+k}^{(3)}v^{k+1}
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\end{align*}
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\textbf{On calcule $E[Z^2]$} \\
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\begin{center}\begin{tabular}{|l|l|l|l|l|l|}
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\hline
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\multicolumn{1}{|l|}{k} & ${}_tp_x^{(\tau)}$ & \multicolumn{1}{l|}{$(B^{(1)})^2\cdot v^{2(k+1)}$} & \multicolumn{1}{l|}{$(B^{(2)})^2\cdot v^{2(k+1)}$} & \multicolumn{1}{l|}{$(B^{(3)})^2\cdot v^{2(k+1)}$} \\ \hline
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0 & 1 & 37703836365,3502 & 235648977283,439 & 9425959091,33754 \\ \hline
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1 & 0,992812194 & 35539481916,6275 & 222121761978,922 & 8884870479,15689 \\ \hline
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2 & 0,9854786805 & 33499370267,3462 & 209371064170,914 & 8374842566,83654 \\ \hline
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3 & 0,9771222992 & 31576369372,5574 & 197352308578,484 & 7894092343,13936 \\ \hline
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4 & 0,9677690657 & 29763756595,869 & 186023478724,181 & 7440939148,96725 \\ \hline
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5 & 0,9563909613 & 28055195207,7189 & 175344970048,243 & 7013798801,92973 \\ \hline
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6 & 0,9450519446 & 26444712232,7448 & 165279451454,655 & 6611178058,18619 \\ \hline
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7 & 0,9329088238 & 24926677568,8046 & 155791734805,028 & 6231669392,20114 \\ \hline
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8 & 0,9208294644 & 23495784304,6513 & 146848651904,071 & 5873946076,16282 \\ \hline
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9 & 0,9069868339 & 22147030167,4534 & 138418938546,584 & 5536757541,86335 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
Avec l'approche DUD, on obtient
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{k} & \multicolumn{1}{l|}{$(B^{(1)})^2\cdot v^{2(k+1)} \cdot {}_tp_x^{(\tau)} \cdot q_x^{(1)}$} & \multicolumn{1}{l|}{$(B^{(2)})^2\cdot v^{2(k+1)} \cdot {}_tp_x^{(\tau)} \cdot q_x^{(2)}$} & \multicolumn{1}{l|}{$(B^{(3)})^2\cdot v^{2(k+1)} \cdot {}_tp_x^{(\tau)} \cdot q_x^{(3)}$} \\ \hline
|
|
0 & 188311886,134414 & 234918936,751815 & 11277236,3087944 \\ \hline
|
|
1 & 176208533,219666 & 219820027,78392 & 12312383,20513 \\ \hline
|
|
2 & 164775878,099411 & 411321322,080527 & 12336555,018151 \\ \hline
|
|
3 & 153915299,33896 & 576604477,576034 & 12292285,205765 \\ \hline
|
|
4 & 143532971,10276 & 897081069,392253 & 12897297,331014 \\ \hline
|
|
5 & 133696252,971125 & 835601581,069528 & 12681455,0168297 \\ \hline
|
|
6 & 124465023,812937 & 933955678,046895 & 11805850,7773972 \\ \hline
|
|
7 & 115806965,995961 & 868987682,063995 & 11563325,6244597 \\ \hline
|
|
8 & 107632359,013123 & 1077943989,17784 & 11285015,4284815 \\ \hline
|
|
9 & 99868557,2862686 & 1125777599,23875 & 11469385,0593954 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
$E[Z^2]$ = 8710146879,1316
|
|
|
|
Avec l'approche FC, on obtient
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{k} & \multicolumn{1}{l|}{$(B^{(1)})^2\cdot v^{2(k+1)} \cdot {}_tp_x^{(\tau)} \cdot q_x^{(1)}$} & \multicolumn{1}{l|}{$(B^{(2)})^2\cdot v^{2(k+1)} \cdot {}_tp_x^{(\tau)} \cdot q_x^{(2)}$} & \multicolumn{1}{l|}{$(B^{(3)})^2\cdot v^{2(k+1)} \cdot {}_tp_x^{(\tau)} \cdot q_x^{(3)}$} \\ \hline
|
|
0 & 188312020,949653 & 234918538,35134 & 11277218,5410028 \\ \hline
|
|
1 & 176208666,459321 & 219819649,11592 & 12312365,0419356 \\ \hline
|
|
2 & 164776033,238299 & 411320830,601987 & 12336535,8925708 \\ \hline
|
|
3 & 153915446,783712 & 576604101,901022 & 12292263,3715776 \\ \hline
|
|
4 & 143533040,295832 & 897081501,848949 & 12897262,7344781 \\ \hline
|
|
5 & 133696318,883425 & 835601993,021407 & 12681422,0606795 \\ \hline
|
|
6 & 124465022,409331 & 933956679,603005 & 11805811,0660543 \\ \hline
|
|
7 & 115806965,76207 & 868988629,290064 & 11563287,79389 \\ \hline
|
|
8 & 107632196,345726 & 1077946468,62323 & 11284956,9175145 \\ \hline
|
|
9 & 99868306,1360683 & 1125780949,071 & 11469313,8536554 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
$E[Z^2]$ = 8710153795,96473
|
|
|
|
\textbf{On calcule $E[Y^2]$} \\
|
|
|
|
Comme on a déjà calculé les quantités requises, on obtient directement
|
|
|
|
\begin{tabular}{|l|l|}
|
|
\hline
|
|
DUD & FC \\ \hline
|
|
\multicolumn{1}{|r|}{301180610,958842} & \multicolumn{1}{r|}{301180559,511373} \\ \hline
|
|
\end{tabular}
|
|
|
|
\textbf{On calcule $E[YZ]$} \\
|
|
|
|
\begin{center}\begin{tabular}{|l|l|}
|
|
\hline
|
|
k & \multicolumn{1}{l|}{$\ddot{a}_{\lcroof{min(k+1;5)}}$} \\ \hline
|
|
0 & 1 \\ \hline
|
|
1 & 1,9708737864 \\ \hline
|
|
2 & 2,9134696955 \\ \hline
|
|
3 & 3,8286113549 \\ \hline
|
|
4 & 4,7170984028 \\ \hline
|
|
5 & 4,7170984028 \\ \hline
|
|
6 & 4,7170984028 \\ \hline
|
|
7 & 4,7170984028 \\ \hline
|
|
8 & 4,7170984028 \\ \hline
|
|
9 & 4,7170984028 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
Avec l'approche DUD, on obtient
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|}
|
|
\hline
|
|
k & $\pi B^{(1)} v^{(t+1)} {}_{t}p_{x}^{(\tau)} q_{x+t}^{(1)}\ddot{a}_{\lcroof{min(k+1;5)}}$ & $\pi B^{(2)} v^{(t+1)} {}_{t}p_{x}^{(\tau)} q_{x+t}^{(2)}\ddot{a}_{\lcroof{min(k+1;5)}}$ & $\pi B^{(3)} v^{(t+1)} {}_{t}p_{x}^{(\tau)} q_{x+t}^{(3)}\ddot{a}_{\lcroof{min(k+1;5)}}$ \\ \hline
|
|
0 & 3606566,30718615 & 1799675,50556917 & 431965,303352579 \\ \hline
|
|
1 & 6850766,44469005 & 3418530,63003612 & 957379,988070364 \\ \hline
|
|
2 & 9754270,09739703 & 9739627,71412775 & 1460578,95253145 \\ \hline
|
|
3 & 12332494,6666226 & 18480220,419926 & 1969843,70483992 \\ \hline
|
|
4 & 14594583,2288744 & 36486458,072186 & 2622821,47270904 \\ \hline
|
|
5 & 14002206,6614794 & 7001103,33073968 & 2656295,1461662 \\ \hline
|
|
6 & 13426468,5451509 & 8059919,90341517 & 2547075,14216523 \\ \hline
|
|
7 & 12867269,0108407 & 7724231,90681933 & 2569593,63696756 \\ \hline
|
|
8 & 12317761,2068153 & 9869044,42103373 & 2582980,18435726 \\ \hline
|
|
9 & 11772126,5571125 & 10616191,3086717 & 2703935,17480047 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
Avec l'approche FC, on obtient
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|}
|
|
\hline
|
|
k & $\pi B^{(1)} v^{(t+1)} {}_{t}p_{x}^{(\tau)} q_{x+t}^{(1)}\ddot{a}_{\lcroof{min(k+1;5)}}$ & $\pi B^{(2)} v^{(t+1)} {}_{t}p_{x}^{(\tau)} q_{x+t}^{(2)}\ddot{a}_{\lcroof{min(k+1;5)}}$ & $\pi B^{(3)} v^{(t+1)} {}_{t}p_{x}^{(\tau)} q_{x+t}^{(3)}\ddot{a}_{\lcroof{min(k+1;5)}}$ \\ \hline
|
|
0 & 3606565,99915052 & 1799675,35185948 & 431965,266458566 \\ \hline
|
|
1 & 6850765,85956838 & 3418530,33806056 & 957379,906300859 \\ \hline
|
|
2 & 9754269,26428808 & 9739626,8822694 & 1460578,82778389 \\ \hline
|
|
3 & 12332493,6133084 & 18480218,8415365 & 1969843,53659622 \\ \hline
|
|
4 & 14594581,9823559 & 36486454,9558898 & 2622821,24869474 \\ \hline
|
|
5 & 14002205,4655556 & 7001102,73277778 & 2656294,91929292 \\ \hline
|
|
6 & 13426467,3984007 & 8059919,2150201 & 2547074,9246204 \\ \hline
|
|
7 & 12867267,9118516 & 7724231,24709526 & 2569593,41749942 \\ \hline
|
|
8 & 12317760,1547594 & 9869043,57812194 & 2582979,96374579 \\ \hline
|
|
9 & 11772125,5516591 & 10616190,4019464 & 2703934,94385827 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
Avec DUD, on a $E[ZY]$ = 245221984,644654
|
|
|
|
Avec FC, on a $E[ZY]$ = 245221963,700326
|
|
|
|
\textbf{La solution est donc}\\
|
|
Avec DUD: $V(L)$ = 8520883520,80113
|
|
|
|
Avec FC: $V(L)$ = 8520890428,07545
|
|
|
|
\subsubsection{Probabilité}
|
|
|
|
\begin{align*}
|
|
P(L<0) = P(K(x)>9) &= {}_{10}p_x^{(\tau)} \\
|
|
&= 0,892272874
|
|
\end{align*}
|
|
|
|
\clearpage
|
|
\section{Exercice 2}
|
|
\label{sec:ex2}
|
|
|
|
\subsection{Partie a}
|
|
\label{sec:ex2pta}
|
|
|
|
On calcule les valeur de ${}_{t}p_{31}^{h}$ et ${}_{t}p_{33}^{f}$.
|
|
|
|
\begin{align*}
|
|
{}_{t}p_{31}^{h} &= \prod_{i=1}^t p_{31+i-1}^{h} \\
|
|
{}_{t}p_{33}^{f} &= \prod_{i=1}^t p_{33+i-1}^{f}
|
|
\end{align*}
|
|
|
|
On rappelle que ${}_{t}q_{x} = 1-{}_{t}p_{x} $, on pourra donc obtenir
|
|
${}_{t}q_{31}^{h}$ et ${}_{t}q_{33}^{f}$.
|
|
|
|
Hommes
|
|
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{t} & \multicolumn{1}{l|}{${}_{t}p_{31}^{h}$} & \multicolumn{1}{l|}{${}_{t}q_{31}^{h}$} \\ \hline
|
|
1 & 0,988 & 0,012 \\ \hline
|
|
2 & 0,97318 & 0,02682 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
Femmes
|
|
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{t} & \multicolumn{1}{l|}{${}_{t}p_{33}^{f}$} & \multicolumn{1}{l|}{${}_{t}q_{33}^{f}$} \\ \hline
|
|
1 & 0,986 & 0,014 \\ \hline
|
|
2 & 0,967266 & 0,032734 \\ \hline
|
|
3 & 0,94308435 & 0,05691565 \\ \hline
|
|
4 & 0,9176210726 & 0,0823789275 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
On peut maintenant calculer $Pr(T(31) \leq 2, T(33) \leq 4)$ avec les
|
|
différentes copules
|
|
|
|
\subsubsection{Copule d'indépendance}
|
|
|
|
\begin{align*}
|
|
Pr(T(31) \leq 2, T(33) \leq 4) &= Pr(T(31) \leq 2) \cdot Pr(T(33) \leq 4) \\
|
|
&= {}_{2}q_{31}^h \cdot {}_{4}q_{33}^f \\
|
|
&= 0,02682 \cdot 0,0823789275 \\
|
|
&= 0,0022094028
|
|
\end{align*}
|
|
|
|
\subsubsection{Copule borne inférieure de Fréchet}
|
|
|
|
\begin{align*}
|
|
Pr(T(31) \leq 2, T(33) \leq 4) &= \max( Pr(T(31) \leq 2) + Pr(T(33) \leq 4) - 1 ; 0) \\
|
|
&= \max(0,02682 + 0,0823789275 - 1 ; 0) \\
|
|
&= 0
|
|
\end{align*}
|
|
|
|
\subsubsection{Copule borne supérieure de Fréchet}
|
|
|
|
\begin{align*}
|
|
Pr(T(31) \leq 2, T(33) \leq 4) &= \min( Pr(T(31) \leq 2) ; Pr(T(33) \leq 4)) \\
|
|
&= \min(0,02682 ; 0,0823789275) \\
|
|
&= 0,02682
|
|
\end{align*}
|
|
|
|
\subsubsection{Copule de Frank}
|
|
|
|
\begin{align*}
|
|
C_{\alpha}^{F} (u_1,u_2) &= -\frac{1}{\alpha}\ln\left(1+\frac{\left(e^{-\alpha\,u_1}-1 \right)\left(e^{-\alpha\,u_2}-1 \right)}{\left(e^{-\alpha}-1 \right)} \right) \\
|
|
C_{\alpha}^{F} ({}_{2}q_{31}^h,{}_{4}q_{33}^f) &=
|
|
-\frac{1}{\alpha}\ln\left(1+\frac{\left(e^{-\alpha\,({}_{2}q_{31}^h)}-1
|
|
\right)\left(e^{-\alpha\,({}_{4}q_{33}^f)}-1
|
|
\right)}{\left(e^{-\alpha}-1 \right)} \right)
|
|
\end{align*}
|
|
|
|
|
|
\begin{center}\begin{tabular}{|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$\alpha$} & \multicolumn{1}{l|}{$C_{\alpha}^{F} (u_1,u_2)$} \\ \hline
|
|
-5 & 9,92094971843724E-005 \\ \hline
|
|
-1 & 0,001357481 \\ \hline
|
|
1 & 0,0033160298 \\ \hline
|
|
5 & 0,0087185154 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
\clearpage
|
|
\subsection{Partie b}
|
|
\label{sec:ex2ptb}
|
|
|
|
On calcule les valeur de ${}_{t|}q_{32}^{h}$ et ${}_{t|}q_{32}^{f}$.
|
|
|
|
Hommes
|
|
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{t} & \multicolumn{1}{l|}{${}_{t}p_{32}^{h}$} & \multicolumn{1}{l|}{${}_{t}q_{32}^{h}$} & ${}_{t|}q_{32}^{h}$ \\ \hline
|
|
0 & 1 & 0 & 0,015 \\ \hline
|
|
1 & 0,985 & 0,015 & 0,01773 \\ \hline
|
|
2 & 0,96727 & 0,03273 & 0,02224721 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
Femmes
|
|
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{t} & \multicolumn{1}{l|}{${}_{t}p_{32}^{f}$} & \multicolumn{1}{l|}{${}_{t}q_{32}^{f}$} & ${}_{t|}q_{32}^{f}$ \\ \hline
|
|
0 & 1 & 0 & 0,012 \\ \hline
|
|
1 & 0,988 & 0,012 & 0,013832 \\ \hline
|
|
2 & 0,974168 & 0,025832 & 0,018509192 \\ \hline
|
|
3 & 0,955658808 & 0,044341192 & 0,0238914702 \\ \hline
|
|
\end{tabular}\end{center}
|
|
\label{}
|
|
|
|
|
|
On décompose la probabilité sous la forme de 4 sections
|
|
\begin{align*}
|
|
P(K(32)=1,K(32)=2) &= P(K(32) \leq 1,K(32) \leq 2) - P(K(32) \leq 1,K(32) \leq 1) \\
|
|
&\quad- P(K(32) \leq 0,K(32) \leq 2) + P(K(32) \leq 0,K(32) \leq 1) \\
|
|
\end{align*}
|
|
Avec les copules, on a la forme
|
|
\begin{align*}
|
|
P(K(32)=1,K(32)=2) &= C({}_{2}q_{32}^{h},{}_{3}q_{32}^{f}) -C({}_{2}q_{32}^{h},{}_{2}q_{32}^{f}) \\
|
|
&\quad -C({}_{1}q_{32}^{h},{}_{3}q_{32}^{f}) +C({}_{1}q_{32}^{h},{}_{2}q_{32}^{f}) \\
|
|
\end{align*}
|
|
|
|
\subsubsection{Copule d'indépendance}
|
|
|
|
\begin{align*}
|
|
P(K(32)=1,K(32)=2) &= P(K(32)=1) \cdot P(K(32)=2) \\
|
|
&= {}_{1|}q_{32}^{h} \cdot {}_{2|}q_{32}^{f} \\
|
|
&= 0,01773 \cdot 0,018509192 \\
|
|
&= 0,000328168
|
|
\end{align*}
|
|
|
|
\subsubsection{Copule borne inférieure de Fréchet}
|
|
|
|
\begin{align*}
|
|
P(K(32)=1,K(32)=2) &= max({}_{2}q_{32}^{h}+{}_{3}q_{32}^{f}-1;0) -max({}_{2}q_{32}^{h}+{}_{2}q_{32}^{f}-1;0) \\
|
|
&\quad -max({}_{1}q_{32}^{h}+{}_{3}q_{32}^{f}-1;0) +max({}_{1}q_{32}^{h}+{}_{2}q_{32}^{f}-1;0) \\
|
|
&=0-0-0+0\\
|
|
&=0
|
|
\end{align*}
|
|
|
|
\subsubsection{Copule borne supérieure de Fréchet}
|
|
\begin{align*}
|
|
P(K(32)=1,K(32)=2) &= min({}_{2}q_{32}^{h};{}_{3}q_{32}^{f}) -min({}_{2}q_{32}^{h};{}_{2}q_{32}^{f}) \\
|
|
&\quad -min({}_{1}q_{32}^{h};{}_{3}q_{32}^{f}) +min({}_{1}q_{32}^{h};{}_{2}q_{32}^{f}) \\
|
|
&= 0,03273 - 0,025832 - 0,015 + 0,015\\
|
|
&= 0,006898
|
|
\end{align*}
|
|
|
|
|
|
\subsubsection{Copule de Frank}
|
|
|
|
\begin{align*}
|
|
P(K(32)=1,K(32)=2) &= C^{F}_{\alpha}({}_{2}q_{32}^{h},{}_{3}q_{32}^{f}) -C^{F}_{\alpha}({}_{2}q_{32}^{h},{}_{2}q_{32}^{f}) \\
|
|
&\quad -C^{F}_{\alpha}({}_{1}q_{32}^{h},{}_{3}q_{32}^{f}) +C^{F}_{\alpha}({}_{1}q_{32}^{h},{}_{2}q_{32}^{f}) \\
|
|
\end{align*}
|
|
|
|
Avec $\alpha=-5$
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$u_1$} & \multicolumn{1}{l|}{$u_2$} & \multicolumn{1}{l|}{$\alpha$} & \multicolumn{1}{l|}{$C^{F}_{\alpha}(u_1,u_2)$} \\ \hline
|
|
0,03273 & 0,044341192 & -5 & 5,98652387723446E-005 \\ \hline
|
|
0,03273 & 0,025832 & -5 & 3,32560691572282E-005 \\ \hline
|
|
0,015 & 0,044341192 & -5 & 2,62254930575751E-005 \\ \hline
|
|
0,015 & 0,025832 & -5 & 1,4568123784245E-005 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
Avec $\alpha=-1$
|
|
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$u_1$} & \multicolumn{1}{l|}{$u_2$} & \multicolumn{1}{l|}{$\alpha$} & \multicolumn{1}{l|}{$C^{F}_{\alpha}(u_1,u_2)$} \\ \hline
|
|
0,03273 & 0,044341192 & -1 & 0,0008775243 \\ \hline
|
|
0,03273 & 0,025832 & -1 & 0,0005065796 \\ \hline
|
|
0,015 & 0,044341192 & -1 & 0,0003986971 \\ \hline
|
|
0,015 & 0,025832 & -1 & 0,0002301376 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
Avec $\alpha=1$
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$u_1$} & \multicolumn{1}{l|}{$u_2$} & \multicolumn{1}{l|}{$\alpha$} & \multicolumn{1}{l|}{$C^{F}_{\alpha}(u_1,u_2)$} \\ \hline
|
|
0,03273 & 0,044341192 & 1 & 0,0022118355 \\ \hline
|
|
0,03273 & 0,025832 & 1 & 0,0012998739 \\ \hline
|
|
0,015 & 0,044341192 & 1 & 0,0010220555 \\ \hline
|
|
0,015 & 0,025832 & 1 & 0,0006007993 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
Avec $\alpha=5$
|
|
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$u_1$} & \multicolumn{1}{l|}{$u_2$} & \multicolumn{1}{l|}{$\alpha$} & \multicolumn{1}{l|}{$C^{F}_{\alpha}(u_1,u_2)$} \\ \hline
|
|
0,03273 & 0,044341192 & 5 & 0,0061376338 \\ \hline
|
|
0,03273 & 0,025832 & 5 & 0,0037174366 \\ \hline
|
|
0,015 & 0,044341192 & 5 & 0,0029142517 \\ \hline
|
|
0,015 & 0,025832 & 5 & 0,0017707098 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
On obtient donc, pour $P(K(x)=1,K(y)=2)$
|
|
|
|
|
|
\begin{center}\begin{tabular}{|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$\alpha$} & \multicolumn{1}{l|}{$P(K(x)=1,K(y)=2)$} \\ \hline
|
|
-5 & 1,49518003417862E-005 \\ \hline
|
|
-1 & 0,0002023853 \\ \hline
|
|
1 & 0,0004907053 \\ \hline
|
|
5 & 0,0012766553 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
\clearpage
|
|
\subsection{Partie c}
|
|
\label{sec:ex2ptc}
|
|
|
|
On a les informations suivantes:
|
|
\begin{align*}
|
|
x&=y=30 \\
|
|
n&=5 \\
|
|
i&=0,05
|
|
\end{align*}
|
|
|
|
Le contrat d'assurance prend la forme suivante
|
|
\begin{align*}
|
|
100000 \termins{xy}{5} + 25000 \termins{\overline{xy}}{5}
|
|
\end{align*}
|
|
|
|
La prime sera alors
|
|
\begin{align*}
|
|
P = \frac{100000 \termins{xy}{5} + 25000 \termins{\overline{xy}}{5}}{\ddot{a}_{xy:\lcroof{5}}}
|
|
\end{align*}
|
|
|
|
On se rappelle la relation suivante (Bowers p.280)
|
|
\begin{align*}
|
|
\ddot{a}_{xy:\lcroof{5}} &= \frac{1-\insend{xy}{5}}{d} \\
|
|
&= \frac{1-(\termins{xy}{5}+\pureend{xy}{5})}{d}
|
|
\end{align*}
|
|
Où $\pureend{xy}{5}$ est une dotation pure sur le statut $(xy)$ survivant au temps 5, c'est à dire lorsque K(x)>4 et K(y)>4.
|
|
|
|
\subsubsection{Calcul des probabilités}
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{${}_{t}p_{30}^h$} & \multicolumn{1}{l|}{${}_{t}q_{30}^h$} & \multicolumn{1}{l|}{t} & \multicolumn{1}{l|}{${}_{t}p_{30}^f$} & \multicolumn{1}{l|}{${}_{t}q_{30}^f$} \\ \hline
|
|
1 & 0 & 0 & 1 & 0 \\ \hline
|
|
0,99 & 0,01 & 1 & 0,992 & 0,008 \\ \hline
|
|
0,97812 & 0,02188 & 2 & 0,983072 & 0,016928 \\ \hline
|
|
0,9634482 & 0,0365518 & 3 & 0,971275136 & 0,028724864 \\ \hline
|
|
0,9461061324 & 0,0538938676 & 4 & 0,9576772841 & 0,0423227159 \\ \hline
|
|
0,9243456914 & 0,0756543086 & 5 & 0,9394814157 & 0,0605185843 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
\subsubsection{Calcul de la fonction de répartition conjointe}
|
|
Nous aurons besoin de ces probabilités pour calculer les éléments de la prime
|
|
|
|
Copule indépendante
|
|
|
|
\begin{center}\begin{tabular}{|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{k} & \multicolumn{1}{l|}{P(K(x)<=k,K(y)<=k)} \\ \hline
|
|
1 & 0,00008 \\ \hline
|
|
2 & 0,0003703846 \\ \hline
|
|
3 & 0,0010499455 \\ \hline
|
|
4 & 0,0022809348 \\ \hline
|
|
5 & 0,0045784917 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
Copule borne inférieure de Fréchet
|
|
|
|
\begin{center}\begin{tabular}{|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{k} & \multicolumn{1}{l|}{P(K(x)<=k,K(y)<=k)} \\ \hline
|
|
1 & 0 \\ \hline
|
|
2 & 0 \\ \hline
|
|
3 & 0 \\ \hline
|
|
4 & 0 \\ \hline
|
|
5 & 0 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
Copule borne supérieure de Fréchet
|
|
|
|
\begin{center}\begin{tabular}{|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{k} & \multicolumn{1}{l|}{P(K(x)<=k,K(y)<=k)} \\ \hline
|
|
1 & 0,008 \\ \hline
|
|
2 & 0,016928 \\ \hline
|
|
3 & 0,028724864 \\ \hline
|
|
4 & 0,0423227159 \\ \hline
|
|
5 & 0,0605185843 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
Copule de Frank
|
|
|
|
$P(K(x)<=k,K(y)<=k)$
|
|
\begin{center}\begin{tabular}{|l|l|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{} & \multicolumn{4}{|c|}{$P(K(x)\leq k,K(y)\leq k)$}\\ \hline
|
|
k/$\alpha$ & -5 & -1 & 1 & 5 \\ \hline
|
|
1 & 2,83882158090462E-006 & 4,69782685808434E-005 & 0,0001254329 & 0,00038543 \\ \hline
|
|
2 & 1,3853295864262E-005 & 0,0002197616 & 0,0005748634 & 0,001700651 \\ \hline
|
|
3 & 4,2015116451646E-005 & 0,0006311739 & 0,0016090912 & 0,0045510223 \\ \hline
|
|
4 & 9,8861759759414E-005 & 0,0013921764 & 0,0034455038 & 0,0092836123 \\ \hline
|
|
5 & 0,000220296 & 0,0028493681 & 0,0067919864 & 0,0172843758 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
\subsubsection{Calcul des éléments de la prime}
|
|
|
|
\textbf{1. Calcul de $\termins{xy}{5}$}
|
|
|
|
\begin{align*}
|
|
\termins{xy}{5} &= \sum_{k=0}^{4} v^{k+1} {}_{k|}q_{xy} \\
|
|
&= \sum_{k=0}^{4} v^{k+1} P(min(K(x),K(y))=k) \\
|
|
&= \sum_{k=0}^{4} v^{k+1} \left[ P(K(x)=k,K(y) \geq k) + P(K(x) \geq k,K(y) =k) - P(K(x) = k,K(y) = k) \right] \\
|
|
&= A\cdot\,v + B\cdot\,v^2 +C\cdot\,v^3 +D\cdot\,v^4 +E\cdot\,v^5
|
|
\end{align*}
|
|
|
|
On sépare les 5 éléments de cette somme en 5 sections illustrées comme
|
|
suit
|
|
|
|
\begin{center}\begin{tabular}{r|lllll}
|
|
$+$ & $\uparrow$& $\uparrow$& $\uparrow$& $\uparrow$& $\uparrow$ \\
|
|
4 & A & B & C & D & E $\rightarrow$ \\
|
|
3 & A & B & C & D & D $\rightarrow$ \\
|
|
2 & A & B & C & C & C $\rightarrow$ \\
|
|
1 & A & B & B & B & B $\rightarrow$ \\
|
|
0 & A & A & A & A & A $\rightarrow$ \\ \hline
|
|
$K(Y)$/$K(X)$ & 0 & 1 & 2 & 3 & 4+ \\
|
|
\end{tabular}\end{center}
|
|
|
|
On a alors la décomposition
|
|
|
|
\begin{align*}
|
|
A &= {}_{}q_{30}^h + {}_{}q_{30}^f - C({}_{}q_{30}^h,{}_{}q_{30}^f) \\
|
|
B &= {}_{2}q_{30}^h + {}_{2}q_{30}^f - C({}_{2}q_{30}^h,{}_{2}q_{30}^f) - A \\
|
|
C &= {}_{3}q_{30}^h + {}_{3}q_{30}^f - C({}_{3}q_{30}^h,{}_{3}q_{30}^f) - (A+B) \\
|
|
D &= {}_{4}q_{30}^h + {}_{4}q_{30}^f - C({}_{4}q_{30}^h,{}_{4}q_{30}^f) - (A+B+C) \\
|
|
E &= {}_{5}q_{30}^h + {}_{5}q_{30}^f - C({}_{5}q_{30}^h,{}_{5}q_{30}^f) - (A+B+C+D)
|
|
\end{align*}
|
|
|
|
\textbf{Copule indépendante}
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{k} & Zone & \multicolumn{1}{l|}{$P(min(K(x),K(y))=k$} & \multicolumn{1}{l|}{$v^{k+1}$} & \multicolumn{1}{l|}{Produit} \\ \hline
|
|
0 & A & 0,01792 & 0,9523809524 & 0,0170666667 \\ \hline
|
|
1 & B & 0,0205176154 & 0,9070294785 & 0,018610082 \\ \hline
|
|
2 & C & 0,0257891032 & 0,8638375985 & 0,0222775969 \\ \hline
|
|
3 & D & 0,0297089301 & 0,8227024748 & 0,0244416104 \\ \hline
|
|
4 & E & 0,0376587526 & 0,7835261665 & 0,0295066181 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
On a alors que
|
|
\begin{align*}
|
|
\termins{xy}{5} = 0,111902574
|
|
\end{align*}
|
|
|
|
|
|
\textbf{Copule borne inférieure de Fréchet}
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{k} & \multicolumn{1}{l|}{Zone} & \multicolumn{1}{l|}{$P(min(K(x),K(y))=k$} & \multicolumn{1}{l|}{$v^{k+1}$} & \multicolumn{1}{l|}{Produit} \\ \hline
|
|
0 & A & 0,018 & 0,9523809524 & 0,0171428571 \\ \hline
|
|
1 & B & 0,020808 & 0,9070294785 & 0,0188734694 \\ \hline
|
|
2 & C & 0,026468664 & 0,8638375985 & 0,0228646271 \\ \hline
|
|
3 & D & 0,0309399195 & 0,8227024748 & 0,0254543483 \\ \hline
|
|
4 & E & 0,0399563094 & 0,7835261665 & 0,031306814 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
On a alors que
|
|
\begin{align*}
|
|
\termins{xy}{5} = 0,115642116
|
|
\end{align*}
|
|
|
|
\textbf{Copule borne supérieure de Fréchet}
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{k} & \multicolumn{1}{l|}{Zone} & \multicolumn{1}{l|}{$P(min(K(x),K(y))=k$} & \multicolumn{1}{l|}{$v^{k+1}$} & \multicolumn{1}{l|}{Produit} \\ \hline
|
|
0 & A & 0,01 & 0,9523809524 & 0,0095238095 \\ \hline
|
|
1 & B & 0,01188 & 0,9070294785 & 0,0107755102 \\ \hline
|
|
2 & C & 0,0146718 & 0,8638375985 & 0,0126740525 \\ \hline
|
|
3 & D & 0,0173420676 & 0,8227024748 & 0,0142673619 \\ \hline
|
|
4 & E & 0,021760441 & 0,7835261665 & 0,017049875 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
On a alors que
|
|
\begin{align*}
|
|
\termins{xy}{5} = 0,0642906091
|
|
\end{align*}
|
|
|
|
\textbf{Copule de Frank}
|
|
|
|
\begin{tabular}{|l|l|l|l|l|l|}
|
|
\hline
|
|
$\alpha$ & \multicolumn{1}{l|}{k} & Zone & \multicolumn{1}{l|}{$P(min(K(x),K(y))=k$} & \multicolumn{1}{l|}{$v^{k+1}$} & \multicolumn{1}{l|}{Produit} \\ \hline
|
|
\multicolumn{1}{|l|}{-5} & 0 & A & 0,0179971612 & 0,9523809524 & 0,0171401535 \\ \hline
|
|
& 1 & B & 0,0207969855 & 0,9070294785 & 0,0188634789 \\ \hline
|
|
& 2 & C & 0,0264405022 & 0,8638375985 & 0,0228402999 \\ \hline
|
|
& 3 & D & 0,0308830729 & 0,8227024748 & 0,0254075805 \\ \hline
|
|
& 4 & E & 0,0398348752 & 0,7835261665 & 0,031211667 \\ \hline
|
|
\multicolumn{1}{|l|}{-1} & 0 & A & 0,0179530217 & 0,9523809524 & 0,0170981159 \\ \hline
|
|
& 1 & B & 0,0206352167 & 0,9070294785 & 0,0187167498 \\ \hline
|
|
& 2 & C & 0,0260572517 & 0,8638375985 & 0,0225092337 \\ \hline
|
|
& 3 & D & 0,030178917 & 0,8227024748 & 0,0248282697 \\ \hline
|
|
& 4 & E & 0,0384991178 & 0,7835261665 & 0,0301650662 \\ \hline
|
|
\multicolumn{1}{|l|}{1} & 0 & A & 0,0178745671 & 0,9523809524 & 0,0170233972 \\ \hline
|
|
& 1 & B & 0,0203585695 & 0,9070294785 & 0,0184658227 \\ \hline
|
|
& 2 & C & 0,0254344362 & 0,8638375985 & 0,0219712223 \\ \hline
|
|
& 3 & D & 0,0291035069 & 0,8227024748 & 0,0239435271 \\ \hline
|
|
& 4 & E & 0,0366098268 & 0,7835261665 & 0,0286847573 \\ \hline
|
|
\multicolumn{1}{|l|}{5} & 0 & A & 0,01761457 & 0,9523809524 & 0,016775781 \\ \hline
|
|
& 1 & B & 0,0194927789 & 0,9070294785 & 0,0176805251 \\ \hline
|
|
& 2 & C & 0,0236182927 & 0,8638375985 & 0,0204023693 \\ \hline
|
|
& 3 & D & 0,0262073295 & 0,8227024748 & 0,0215608348 \\ \hline
|
|
& 4 & E & 0,031955546 & 0,7835261665 & 0,0250380064 \\ \hline
|
|
\end{tabular}
|
|
|
|
On a alors que
|
|
\begin{align*}
|
|
\alpha = -5 \Rightarrow \termins{xy}{5} &= 0,1154631799\\
|
|
\alpha = -1 \Rightarrow \termins{xy}{5} &= 0,1133174353\\
|
|
\alpha = 1 \Rightarrow \termins{xy}{5} &= 0,1100887266\\
|
|
\alpha = 5 \Rightarrow \termins{xy}{5} &= 0,1014575166
|
|
\end{align*}
|
|
|
|
\textbf{2. Calcul de $\termins{\overline{xy}}{5}$}
|
|
|
|
\begin{align*}
|
|
\termins{\overline{xy}}{5} &= \sum_{k=0}^{4} v^{k+1} {}_{k|}q_{\overline{xy}} \\
|
|
&= \sum_{k=0}^{4} v^{k+1} \left[F_{K(\overline{xy})}(k+1) - F_{K(\overline{xy})}(k)\right] \\
|
|
&= \sum_{k=0}^{4} v^{k+1} \left[C({}_{k+1}q_{30}^{h},{}_{k+1}q_{30}^{f}) - C({}_{k}q_{30}^{h},{}_{k}q_{30}^{f}) \right] \\
|
|
\end{align*}
|
|
|
|
\textbf{Copule indépendante}
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{${}_{k|}q_{\overline{xy}}$} & \multicolumn{1}{l|}{$v^{k+1}$} & \multicolumn{1}{l|}{Produit} \\ \hline
|
|
0,00008 & 0,9523809524 & 7,61904761904763E-005 \\ \hline
|
|
0,0002903846 & 0,9070294785 & 0,0002633874 \\ \hline
|
|
0,0006795608 & 0,8638375985 & 0,0005870302 \\ \hline
|
|
0,0012309894 & 0,8227024748 & 0,001012738 \\ \hline
|
|
0,0022975568 & 0,7835261665 & 0,0018001959 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
On a alors
|
|
\begin{align*}
|
|
\termins{\overline{xy}}{5} = 0,003739542
|
|
\end{align*}
|
|
|
|
\textbf{Copule borne inférieure de Fréchet}
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{${}_{k|}q_{\overline{xy}}$} & \multicolumn{1}{l|}{$v^{k+1}$} & \multicolumn{1}{l|}{Produit} \\ \hline
|
|
0 & 0,9523809524 & 0 \\ \hline
|
|
0 & 0,9070294785 & 0 \\ \hline
|
|
0 & 0,8638375985 & 0 \\ \hline
|
|
0 & 0,8227024748 & 0 \\ \hline
|
|
0 & 0,7835261665 & 0 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
On a alors
|
|
\begin{align*}
|
|
\termins{\overline{xy}}{5} = 0
|
|
\end{align*}
|
|
|
|
\textbf{Copule borne supérieure de Fréchet}
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{${}_{k|}q_{\overline{xy}}$} & \multicolumn{1}{l|}{$v^{k+1}$} & \multicolumn{1}{l|}{Produit} \\ \hline
|
|
0,008 & 0,9523809524 & 0,0076190476 \\ \hline
|
|
0,008928 & 0,9070294785 & 0,0080979592 \\ \hline
|
|
0,011796864 & 0,8638375985 & 0,0101905747 \\ \hline
|
|
0,0135978519 & 0,8227024748 & 0,0111869864 \\ \hline
|
|
0,0181958684 & 0,7835261665 & 0,014256939 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
On a alors
|
|
\begin{align*}
|
|
\termins{\overline{xy}}{5} = 0,0513515069
|
|
\end{align*}
|
|
|
|
\textbf{Copule de Frank}
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|}
|
|
\hline
|
|
$\alpha$ & \multicolumn{1}{l|}{${}_{k|}q_{\overline{xy}}$} & \multicolumn{1}{l|}{$v^{k+1}$} & \multicolumn{1}{l|}{Produit} \\ \hline
|
|
\multicolumn{1}{|l|}{-5} & 2,83882158090462E-006 & 0,9523809524 & 2,70363960086155E-006 \\ \hline
|
|
& 1,10144742833573E-005 & 0,9070294785 & 9,99045286472321E-006 \\ \hline
|
|
& 2,8161820587384E-005 & 0,8638375985 & 2,43272394664801E-005 \\ \hline
|
|
& 5,6846643307768E-005 & 0,8227024748 & 4,67678741329121E-005 \\ \hline
|
|
& 0,0001214343 & 0,7835261665 & 9,51469145818031E-005 \\ \hline
|
|
& \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} \\ \hline
|
|
\multicolumn{1}{|l|}{-1} & 4,69782685808434E-005 & 0,9523809524 & 4,47412081722318E-005 \\ \hline
|
|
& 0,0001727833 & 0,9070294785 & 0,0001567195 \\ \hline
|
|
& 0,0004114123 & 0,8638375985 & 0,0003553935 \\ \hline
|
|
& 0,0007610025 & 0,8227024748 & 0,0006260787 \\ \hline
|
|
& 0,0014571917 & 0,7835261665 & 0,0011417478 \\ \hline
|
|
& \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} \\ \hline
|
|
\multicolumn{1}{|l|}{1} & 0,0001254329 & 0,9523809524 & 0,0001194599 \\ \hline
|
|
& 0,0004494305 & 0,9070294785 & 0,0004076467 \\ \hline
|
|
& 0,0010342278 & 0,8638375985 & 0,0008934048 \\ \hline
|
|
& 0,0018364126 & 0,8227024748 & 0,0015108212 \\ \hline
|
|
& 0,0033464826 & 0,7835261665 & 0,0026220567 \\ \hline
|
|
& \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} \\ \hline
|
|
\multicolumn{1}{|l|}{5} & 0,00038543 & 0,9523809524 & 0,0003670762 \\ \hline
|
|
& 0,0013152211 & 0,9070294785 & 0,0011929443 \\ \hline
|
|
& 0,0028503713 & 0,8638375985 & 0,0024622579 \\ \hline
|
|
& 0,00473259 & 0,8227024748 & 0,0038935135 \\ \hline
|
|
& 0,0080007635 & 0,7835261665 & 0,0062688075 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
On a alors que
|
|
\begin{align*}
|
|
\alpha = -5 \Rightarrow \termins{\overline{xy}}{5} &= 0,0001789361 \\
|
|
\alpha = -1 \Rightarrow \termins{\overline{xy}}{5} &= 0,0023246807 \\
|
|
\alpha = 1 \Rightarrow \termins{\overline{xy}}{5} &= 0,0055533894 \\
|
|
\alpha = 5 \Rightarrow \termins{\overline{xy}}{5} &= 0,0141845994
|
|
\end{align*}
|
|
|
|
\textbf{3. Calcul de la dotation pure $\pureend{xy}{5}$}
|
|
|
|
Si on revient au schéma avec les sections, on a que
|
|
\begin{align*}
|
|
P(Kx>4,Ky>4) &= 1-(A+B+C+D+E) \\
|
|
&= 1 - ({}_{5}q_{30}^{h} + {}_{5}q_{30}^{f} - C({}_{5}q_{30}^{h} , {}_{5}q_{30}^{f})
|
|
\end{align*}
|
|
|
|
alors
|
|
\begin{align*}
|
|
\pureend{xy}{5} = \left[1 - ({}_{5}q_{30}^{h} + {}_{5}q_{30}^{f} - C({}_{5}q_{30}^{h} , {}_{5}q_{30}^{f})\right]v^5
|
|
\end{align*}
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{Copule} &\multicolumn{1}{|l|}{$P(K(x)>4,K(y)>4)$} & \multicolumn{1}{l|}{$v^5$} & \multicolumn{1}{l|}{$\pureend{xy}{5}$} \\ \hline
|
|
Copule d'indépendance & 0,8684055987 & 0,7835261665 & 0,6804185097 \\ \hline
|
|
Copule borne inférieure de Fréchet & 0,8638271071 & 0,7835261665 & 0,6768311417 \\ \hline
|
|
Copule borne supérieure de Fréchet & 0,9243456914 & 0,7835261665 & 0,724249036 \\ \hline
|
|
Copule de Frank ($\alpha=-5$) & 0,8640474031 & 0,7835261665 & 0,6770037494 \\ \hline
|
|
Copule de Frank ($\alpha=-1$) & 0,8666764752 & 0,7835261665 & 0,6790636961 \\ \hline
|
|
Copule de Frank ($\alpha=1$) & 0,8706190935 & 0,7835261665 & 0,6821528408 \\ \hline
|
|
Copule de Frank ($\alpha=5$) & 0,8811114829 & 0,7835261665 & 0,6903739024 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
\textbf{4. Calcul de la prime $P$}
|
|
|
|
On peut maintenant calculer la prime
|
|
|
|
\begin{center}\begin{tabular}{|l|l|}
|
|
\hline
|
|
Copule d'indépendance & 2587,2690654571 \\ \hline
|
|
Copule borne inférieure de Fréchet & 2653,5218382164 \\ \hline
|
|
Copule borne supérieure de Fréchet & 1736,8669607501 \\ \hline
|
|
Copule de Frank ($\alpha=-5$) & 2650,3616180178 \\ \hline
|
|
Copule de Frank ($\alpha=-1$) & 2612,3555902388 \\ \hline
|
|
Copule de Frank ($\alpha=1$) & 2555,0982581424 \\ \hline
|
|
Copule de Frank ($\alpha=5$) & 2401,9833198864 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
\subsection{Partie d}
|
|
\label{sec:ex2ptd}
|
|
|
|
On a $x=y=30$ et $n=10$. La rente peut être représentée sous la forme
|
|
\begin{align*}
|
|
PUN = E[y] &= 75000 \ddot{a}_{xy:\lcroof{10}} + 50000 (\ddot{a}_{x:\lcroof{10}}-\ddot{a}_{xy:\lcroof{10}}) + 35000 (\ddot{a}_{y:\lcroof{10}}-\ddot{a}_{xy:\lcroof{10}}) \\
|
|
&= 50000 \ddot{a}_{x:\lcroof{10}} + 35000 \ddot{a}_{y:\lcroof{10}} - 10000 \ddot{a}_{xy:\lcroof{10}}
|
|
\end{align*}
|
|
|
|
On calcule d'abord $\ddot{a}_{x:\lcroof{10}}$ et $\ddot{a}_{y:\lcroof{10}}$
|
|
|
|
\begin{align*}
|
|
\ddot{a}_{x:\lcroof{10}} &= \sum_{k=0}^9 v^k {}_{k}p_x^{(h)}\\
|
|
\ddot{a}_{y:\lcroof{10}} &= \sum_{k=0}^9 v^k {}_{k}p_y^{(f)}
|
|
\end{align*}
|
|
|
|
\begin{center}\begin{tabular}{|r|r|r|r|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$k$} & \multicolumn{1}{l|}{$v^k$} & \multicolumn{1}{l|}{${}_{k}p_{30}^{(h)}$} & \multicolumn{1}{l|}{${}_{k}p_{30}^{(f)}$} \\ \hline
|
|
0 & 1 & 1 & 1 \\ \hline
|
|
1 & 0,9523809524 & 0,99 & 0,992 \\ \hline
|
|
2 & 0,9070294785 & 0,97812 & 0,983072 \\ \hline
|
|
3 & 0,8638375985 & 0,9634482 & 0,971275136 \\ \hline
|
|
4 & 0,8227024748 & 0,9461061324 & 0,9576772841 \\ \hline
|
|
5 & 0,7835261665 & 0,9243456914 & 0,9394814157 \\ \hline
|
|
6 & 0,7462153966 & 0,8966153206 & 0,9159943803 \\ \hline
|
|
7 & 0,7106813301 & 0,867027015 & 0,891262532 \\ \hline
|
|
8 & 0,676839362 & 0,8366810695 & 0,8645246561 \\ \hline
|
|
9 & 0,6446089162 & 0,8073972321 & 0,8368598671 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
On obtient:
|
|
\begin{align*}
|
|
\ddot{a}_{x:\lcroof{10}} &= 7,5369186937 \\
|
|
\ddot{a}_{y:\lcroof{10}} &= 7,6409773132
|
|
\end{align*}
|
|
|
|
On calcule maintenant $\ddot{a}_{xy:\lcroof{10}}$
|
|
|
|
\begin{align*}
|
|
\ddot{a}_{xy:\lcroof{10}} &= \sum_{k=0}^9 v^k {}_{k}p_{xy} \\
|
|
&= \sum_{k=0}^9 v^k (1- {}_{k}q_{xy}) \\
|
|
&= \sum_{k=0}^9 v^k \left( 1-({}_{k}q_{x} + {}_{k}q_{y} - C({}_{k}q_{x},{}_{k}q_{y})) \right)
|
|
\end{align*}
|
|
|
|
On évalue d'abord $1-({}_{k}q_{x} + {}_{k}q_{y} - C({}_{k}q_{x},{}_{k}q_{y}))$ pour différentes copules
|
|
\begin{center}\begin{tabular}{|l|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{k} & \multicolumn{1}{l|}{avec indépendance} & \multicolumn{1}{l|}{avec Borne Inf. Fréchet} & \multicolumn{1}{l|}{avec Borne Sup. Fréchet} \\ \hline
|
|
0 & 1 & 1 & 1 \\ \hline
|
|
1 & 0,98208 & 0,982 & 0,99 \\ \hline
|
|
2 & 0,9615623846 & 0,961192 & 0,97812 \\ \hline
|
|
3 & 0,9357732815 & 0,934723336 & 0,9634482 \\ \hline
|
|
4 & 0,9060643513 & 0,9037834165 & 0,9461061324 \\ \hline
|
|
5 & 0,8684055987 & 0,8638271071 & 0,9243456914 \\ \hline
|
|
6 & 0,821294595 & 0,8126097009 & 0,8966153206 \\ \hline
|
|
7 & 0,7727486928 & 0,7582895471 & 0,867027015 \\ \hline
|
|
8 & 0,7233314139 & 0,7012057256 & 0,8366810695 \\ \hline
|
|
9 & 0,6756783403 & 0,6442570992 & 0,8073972321 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
\begin{center}\begin{tabular}{|l|l|l|l|l|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{k} & avec $C_{\alpha=-5}^F$ & avec $C_{\alpha=-1}^F$ & avec $C_{\alpha=1}^F$ & avec $C_{\alpha=5}^F$ \\ \hline
|
|
0 & 1 & 1 & 1 & 1 \\ \hline
|
|
1 & 0,9820028388 & 0,9820469783 & 0,9821254329 & 0,98238543 \\ \hline
|
|
2 & 0,9612058533 & 0,9614117616 & 0,9617668634 & 0,962892651 \\ \hline
|
|
3 & 0,9347653511 & 0,9353545099 & 0,9363324272 & 0,9392743583 \\ \hline
|
|
4 & 0,9038822783 & 0,9051755929 & 0,9072289203 & 0,9130670288 \\ \hline
|
|
5 & 0,8640474031 & 0,8666764752 & 0,8706190935 & 0,8811114829 \\ \hline
|
|
6 & 0,8130884932 & 0,818149309 & 0,825208447 & 0,8426298621 \\ \hline
|
|
7 & 0,7592127838 & 0,7677521898 & 0,7787933563 & 0,8041884324 \\ \hline
|
|
8 & 0,702858518 & 0,7160742106 & 0,7318727321 & 0,7659257251 \\ \hline
|
|
9 & 0,6470083967 & 0,6659249819 & 0,6868706729 & 0,7295307217 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
On obtient les valeurs pour $\ddot{a}_{xy:\lcroof{10}}$
|
|
|
|
\begin{center}\begin{tabular}{|l|l|}
|
|
\hline
|
|
copule & \multicolumn{1}{l|}{$\ddot{a}_{xy:\lcroof{10}}$} \\ \hline
|
|
avec indépendance & 7,1288439482 \\ \hline
|
|
avec Borne Inf. Fréchet & 7,0700743313 \\ \hline
|
|
avec Borne Sup. Fréchet & 7,5369186937 \\ \hline
|
|
avec $C_{\alpha=-5}^F$ & 7,0742854312 \\ \hline
|
|
avec $C_{\alpha=-1}^F$ & 7,1091310866 \\ \hline
|
|
avec $C_{\alpha=1}^F$ & 7,1524603059 \\ \hline
|
|
avec $C_{\alpha=5}^F$ & 7,2508898718 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
La solution est donc
|
|
|
|
\begin{center}\begin{tabular}{|l|l|}
|
|
\hline
|
|
copule & PUN \\ \hline
|
|
avec indépendance & 572991,701166855 \\ \hline
|
|
avec Borne Inf. Fréchet & 573579,397335995 \\ \hline
|
|
avec Borne Sup. Fréchet & 568910,953711506 \\ \hline
|
|
avec $C_{\alpha=-5}^F$ & 573537,286336841 \\ \hline
|
|
avec $C_{\alpha=-1}^F$ & 573188,829782867 \\ \hline
|
|
avec $C_{\alpha=1}^F$ & 572755,53759031 \\ \hline
|
|
avec $C_{\alpha=5}^F$ & 571771,241931411 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
\clearpage
|
|
\section{Exercice 3}
|
|
\label{sec:ex3}
|
|
|
|
\subsection{Partie a}
|
|
\label{sec:ex3pta}
|
|
On calcule, à partir des hypothèses du problème, une table pour les fonctions de survie de $T^{*}(x=33)$
|
|
\begin{center}\begin{tabular}{|r|r|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$T^{*}(33)^h$} & \multicolumn{1}{l|}{$S(T^{*}(33)^h)$} \\ \hline
|
|
1 & 0,982 \\ \hline
|
|
2 & 0,959414 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
, $T^{*}(y=32)$
|
|
\begin{center}\begin{tabular}{|r|r|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$T^{*}(32)^f$} & \multicolumn{1}{l|}{$S(T^{*}(32)^f)$} \\ \hline
|
|
1 & 0,988 \\ \hline
|
|
2 & 0,974168 \\ \hline
|
|
3 & 0,955658808 \\ \hline
|
|
4 & 0,9317673378 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
et $T(W)$, où W représente le temps d'arrivée du choc commun
|
|
\begin{center}\begin{tabular}{|r|r|r|r|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$T(W)$} & \multicolumn{1}{l|}{$S(T(W),\alpha=0.05)$} & \multicolumn{1}{l|}{$S(T(W),\alpha=0.2)$} & \multicolumn{1}{l|}{$S(T(W),\alpha=0.001)$} \\ \hline
|
|
1 & 0,9512294245 & 0,8187307531 & 0,9990004998 \\ \hline
|
|
2 & 0,904837418 & 0,670320046 & 0,9980019987 \\ \hline
|
|
3 & 0,8607079764 & 0,5488116361 & 0,9970044955 \\ \hline
|
|
4 & 0,8187307531 & 0,4493289641 & 0,9960079893 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
On développe la fonction de survie conjointe en fonction de ces fonctions de survie en utilisant l'équation
|
|
\begin{align*}
|
|
S_{T(x)T(y)}(s,t) = S_{T^{*}(x)}(s)S_{T^{*}(y)}(t)S_{W}(max(s,t))
|
|
\end{align*}
|
|
|
|
On obtient alors une expression pour la probabilité recherchée en fonction de ces fonctions de survie
|
|
\begin{align*}
|
|
P(1 \leq T(x) \leq 2 ; 2 \leq T(y) \leq 4) &= S_{T(x)T(y)}(1,2) - S_{T(x)T(y)}(1,4) - S_{T(x)T(y)}(2,2) + S_{T(x)T(y)}(2,4) \\
|
|
&= S_{T^{*}(x)}(1)S_{T^{*}(y)}(2)S_{W}(2) - S_{T^{*}(x)}(1)S_{T^{*}(y)}(4)S_{W}(4) \\
|
|
&\quad- S_{T^{*}(x)}(2)S_{T^{*}(y)}(2)S_{W}(2) + S_{T^{*}(x)}(2)S_{T^{*}(y)}(4)S_{W}(4)
|
|
\end{align*}
|
|
|
|
On peut ensuite évaluer numériquement l'expression et obtenir la probabilité recherchée
|
|
|
|
\begin{center}\begin{tabular}{|l|r|r|r|}
|
|
\hline
|
|
$\alpha$ & 0,05 & 0,2 & 0,001 \\ \hline
|
|
$S_{T(x)T(y)}(1,2)$ & 0,865597312 & 0,6412502605 & 0,954721622 \\ \hline
|
|
$S_{T(x)T(y)}(1,4)$ & 0,7491349758 & 0,4111339917 & 0,9113428538 \\ \hline
|
|
$S_{T(x)T(y)}(2,2)$ & 0,8456885738 & 0,6265015045 & 0,9327630247 \\ \hline
|
|
$S_{T(x)T(y)}(2,4)$ & 0,7319048714 & 0,4016779099 & 0,8903819682 \\ \hline
|
|
$P(1 \leq T(x) \leq 2 ; 2 \leq T(y) \leq 4)$ & 0,0026786337 & 0,0052926742 & 0,0009977117 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
\subsection{Partie b}
|
|
\label{sec:ex3ptb}
|
|
|
|
Ce problème est similaire à la partie a)
|
|
|
|
Pour $S(T^{*}(30)^h)$, on a:
|
|
\begin{center}\begin{tabular}{|r|r|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$T^{*}(30)^h$} & \multicolumn{1}{l|}{$S(T^{*}(30)^h)$} \\ \hline
|
|
1 & 0,99 \\ \hline
|
|
2 & 0,97812 \\ \hline
|
|
3 & 0,9634482 \\ \hline
|
|
4 & 0,9461061324 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
Et pour $S(T^{*}(31)^f)$:
|
|
\begin{center}\begin{tabular}{|r|r|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$T^{*}(31)^f$} & \multicolumn{1}{l|}{$S(T^{*}(31)^f)$} \\ \hline
|
|
1 & 0,991 \\ \hline
|
|
2 & 0,979108 \\ \hline
|
|
3 & 0,965400488 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
Comme $W$ suit une distribution exponentielle (loi sans mémoire), on peut réutiliser le tableau précédent.
|
|
La probabilité $P(K(x)=3 ; K(y)=2)$ s'exprime sous la forme de fonctions de survie:
|
|
\begin{align*}
|
|
P(K(x)=3 ; K(y)=2) &= S_{T(x)T(y)}(3,2) - S_{T(x)T(y)}(4,2) - S_{T(x)T(y)}(3,3) + S_{T(x)T(y)}(4,3) \\
|
|
&= S_{T^{*}(x)}(3)S_{T^{*}(y)}(2)S_{W}(3) - S_{T^{*}(x)}(4)S_{T^{*}(y)}(2)S_{W}(4) \\
|
|
&\quad - S_{T^{*}(x)}(3)S_{T^{*}(y)}(3)S_{W}(3) + S_{T^{*}(x)}(4)S_{T^{*}(y)}(3)S_{W}(4)
|
|
\end{align*}
|
|
|
|
On obtient donc
|
|
\begin{center}
|
|
\begin{tabular}{|l|r|r|r|}
|
|
\hline
|
|
$\alpha$ & 0,05 & 0,2 & 0,001 \\ \hline
|
|
$S_{T(x)T(y)}(3,2)$ & 0,8119229108 & 0,5177049049 & 0,9404941214 \\ \hline
|
|
$S_{T(x)T(y)}(4,2)$ & 0,7584231138 & 0,41623143 & 0,9226421236 \\ \hline
|
|
$S_{T(x)T(y)}(3,3)$ & 0,80055599 & 0,5104570362 & 0,9273272037 \\ \hline
|
|
$S_{T(x)T(y)}(4,3)$ & 0,7478051902 & 0,4104041899 & 0,9097251339 \\ \hline
|
|
$P(K(x)=3 ; K(y)=2)$ & 0,0007489972 & 0,0014206286 & 0,000249928 \\ \hline
|
|
\end{tabular}
|
|
\end{center}
|
|
|
|
\subsection{Partie c}
|
|
\label{sec:ex3ptc}
|
|
|
|
On décrit la valeur de ce contrat sous la forme
|
|
\begin{align*}
|
|
PUN = 100000 \lcterm{A}{xy}{10} + 75000 \lcterm{A}{\overline{xy}}{10} - 135000 A_{(K(x)=K(y)):\lcroof{10}}
|
|
\end{align*}
|
|
|
|
On évalue
|
|
\begin{align*}
|
|
\lcterm{A}{xy}{10} &= \sum_{k=0}^9 v^{k+1} {}_{k|}q_{xy} \\
|
|
&= \sum_{k=0}^9 v^{k+1} \left[S_{T(x)T(y)}{(k,k)} - S_{T(x)T(y)}{(k+1,k+1)}\right] \\
|
|
\lcterm{A}{\overline{xy}}{10} &= \sum_{k=0}^9 v^{k+1} {}_{k|}q_{\overline{xy}} \\
|
|
&= \sum_{k=0}^9 v^{k+1} \bigg\{\left[S_{T(x)}{(k)} + S_{T(y)}{(k)} - S_{T(x)T(y)}{(k,k)} \right] \\
|
|
&\quad - \left[S_{T(x)}{(k+1)} + S_{T(y)}{(k+1)} - S_{T(x)T(y)}{(k+1,k+1)} \right]\bigg\} \\
|
|
\lcterm{A}{(T(x)=T(y))}{10} &= \sum_{k=0}^9 v^{k+1} P(T(x)=k,T(y)=k) \\
|
|
&= \sum_{k=0}^9 v^{k+1} \left[S_{T(x)T(y)}{(k,k)} - S_{T(x)T(y)}{(k+1,k)} - S_{T(x)T(y)}{(k,k+1)} + S_{T(x)T(y)}{(k+1,k+1)} \right]
|
|
\end{align*}
|
|
|
|
avec le choc commun, on a
|
|
|
|
\begin{align*}
|
|
S_{T(x)T(y)}{(k,k)} &= S_{T(x)*}{(k)} \cdot S_{T(y)*}{(k)} \cdot S_{W}{(k)} \\
|
|
S_{T(x)T(y)}{(k,l)} &= S_{T(x)*}{(k)} \cdot S_{T(y)*}{(l)} \cdot S_{W}{(\max(k,l))} \\
|
|
S_{T(x)}{(k)} &= S_{T(x)*}{(k)} \cdot S_{W}{(k)} \\
|
|
S_{T(y)}{(k)} &= S_{T(y)*}{(k)} \cdot S_{W}{(k)}
|
|
\end{align*}
|
|
|
|
Résultats numériques
|
|
|
|
\begin{center}\begin{tabular}{|r|r|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$T^{*}(30)^{(h)}$} & \multicolumn{1}{l|}{$S(T^{*}(30)^{(h)})$} \\ \hline
|
|
0 & 1 \\ \hline
|
|
1 & 0,99 \\ \hline
|
|
2 & 0,97812 \\ \hline
|
|
3 & 0,9634482 \\ \hline
|
|
4 & 0,9461061324 \\ \hline
|
|
5 & 0,9243456914 \\ \hline
|
|
6 & 0,8966153206 \\ \hline
|
|
7 & 0,867027015 \\ \hline
|
|
8 & 0,8366810695 \\ \hline
|
|
9 & 0,8073972321 \\ \hline
|
|
10 & 0,779138329 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
\begin{center}\begin{tabular}{|r|r|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$T^{*}(30)^{(f)}$} & \multicolumn{1}{l|}{$S(T^{*}(30)^{(f)})$} \\ \hline
|
|
0 & 1 \\ \hline
|
|
1 & 0,992 \\ \hline
|
|
2 & 0,983072 \\ \hline
|
|
3 & 0,971275136 \\ \hline
|
|
4 & 0,9576772841 \\ \hline
|
|
5 & 0,9394814157 \\ \hline
|
|
6 & 0,9159943803 \\ \hline
|
|
7 & 0,891262532 \\ \hline
|
|
8 & 0,8645246561 \\ \hline
|
|
9 & 0,8368598671 \\ \hline
|
|
10 & 0,8092434915 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
\begin{center}\begin{tabular}{|r|r|r|r|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$k$} & \multicolumn{1}{l|}{$S_{T(x)}{(k)};\alpha=0,05$} & \multicolumn{1}{l|}{$S_{T(x)}{(k)};\alpha=0,2$} & \multicolumn{1}{l|}{$S_{T(x)}{(k)};\alpha=0,001$} \\ \hline
|
|
0 & 1 & 1 & 1 \\ \hline
|
|
1 & 0,9417171303 & 0,8105434455 & 0,9890104948 \\ \hline
|
|
2 & 0,8850395753 & 0,6556534434 & 0,9761657149 \\ \hline
|
|
3 & 0,8292475506 & 0,5287515829 & 0,9605621866 \\ \hline
|
|
4 & 0,7746061863 & 0,4251128884 & 0,9423292666 \\ \hline
|
|
5 & 0,7198811483 & 0,3400477764 & 0,919735498 \\ \hline
|
|
6 & 0,6642289665 & 0,2700553449 & 0,8912517355 \\ \hline
|
|
7 & 0,610983611 & 0,2138062296 & 0,8609790186 \\ \hline
|
|
8 & 0,560844093 & 0,1689229946 & 0,8300143235 \\ \hline
|
|
9 & 0,5148192047 & 0,1334618648 & 0,8001632587 \\ \hline
|
|
10 & 0,4725712847 & 0,1054449064 & 0,771385773 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
\begin{center}\begin{tabular}{|r|r|r|r|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$k$} & \multicolumn{1}{l|}{$S_{T(y)}{(k)};\alpha=0,05$} & \multicolumn{1}{l|}{$S_{T(y)}{(k)};\alpha=0,2$} & \multicolumn{1}{l|}{$S_{T(y)}{(k)};\alpha=0,001$} \\ \hline
|
|
0 & 1 & 1 & 1 \\ \hline
|
|
1 & 0,9436195891 & 0,8121809071 & 0,9910084958 \\ \hline
|
|
2 & 0,8895203302 & 0,6589728683 & 0,9811078208 \\ \hline
|
|
3 & 0,8359842569 & 0,5330470965 & 0,968365677 \\ \hline
|
|
4 & 0,784079844 & 0,430312142 & 0,9538542262 \\ \hline
|
|
5 & 0,7316688622 & 0,3456158982 & 0,9347957326 \\ \hline
|
|
6 & 0,678585327 & 0,2758922055 & 0,910514869 \\ \hline
|
|
7 & 0,6280620911 & 0,2197826345 & 0,8850454794 \\ \hline
|
|
8 & 0,5795082073 & 0,1745445178 & 0,85763605 \\ \hline
|
|
9 & 0,5336054102 & 0,1383320056 & 0,8293619197 \\ \hline
|
|
10 & 0,4908309887 & 0,1095191971 & 0,8011913842 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
\begin{center}\begin{tabular}{|r|r|r|r|r|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$k^{(h)}$} & \multicolumn{1}{l|}{$k^{(f)}$} & \multicolumn{1}{l|}{$S_{T(x)T(y)}{(k^{(h)},k^{(f)})};\alpha=0,05$} & \multicolumn{1}{l|}{$S_{T(x)T(y)}{(k^{(h)},k^{(f)})};\alpha=0,2$} & \multicolumn{1}{l|}{$S_{T(x)T(y)}{(k^{(h)},k^{(f)})};\alpha=0,001$} \\ \hline
|
|
0 & 0 & 1 & 1 & 1 \\ \hline
|
|
1 & 1 & 0,9341833932 & 0,804059098 & 0,9810984109 \\ \hline
|
|
2 & 2 & 0,8700576254 & 0,6445545419 & 0,9596411817 \\ \hline
|
|
3 & 3 & 0,8054275275 & 0,5135632656 & 0,9329701684 \\ \hline
|
|
4 & 4 & 0,7418227487 & 0,4071209564 & 0,9024473328 \\ \hline
|
|
5 & 5 & 0,6763149603 & 0,3194685664 & 0,8640744077 \\ \hline
|
|
6 & 6 & 0,6084300005 & 0,2473691783 & 0,8163815812 \\ \hline
|
|
7 & 7 & 0,5445468001 & 0,1905574815 & 0,7673583402 \\ \hline
|
|
8 & 8 & 0,4848635466 & 0,1460380938 & 0,7175678476 \\ \hline
|
|
9 & 9 & 0,4308315312 & 0,1116888784 & 0,6696245183 \\ \hline
|
|
10 & 10 & 0,3824252364 & 0,0853306042 & 0,6242389162 \\ \hline
|
|
1 & 0 & 0,9417171303 & 0,8105434455 & 0,9890104948 \\ \hline
|
|
2 & 1 & 0,8779592587 & 0,6504082159 & 0,9683563892 \\ \hline
|
|
3 & 2 & 0,8152100481 & 0,5198008761 & 0,9443017899 \\ \hline
|
|
4 & 3 & 0,7523557289 & 0,4129015785 & 0,9152609866 \\ \hline
|
|
5 & 4 & 0,6894138229 & 0,325656031 & 0,8808097938 \\ \hline
|
|
6 & 5 & 0,6240307698 & 0,2537119777 & 0,8373144422 \\ \hline
|
|
7 & 6 & 0,5596575541 & 0,1958453048 & 0,7886519426 \\ \hline
|
|
8 & 7 & 0,4998593264 & 0,1505547359 & 0,7397606676 \\ \hline
|
|
9 & 8 & 0,4450738959 & 0,1153810728 & 0,691760866 \\ \hline
|
|
10 & 9 & 0,3954759425 & 0,0882426104 & 0,6455417955 \\ \hline
|
|
0 & 1 & 0,9436195891 & 0,8121809071 & 0,9910084958 \\ \hline
|
|
1 & 2 & 0,8806251269 & 0,6523831396 & 0,9712967426 \\ \hline
|
|
2 & 3 & 0,8176929213 & 0,521384026 & 0,947177836 \\ \hline
|
|
3 & 4 & 0,7554203144 & 0,4145834587 & 0,9189891373 \\ \hline
|
|
4 & 5 & 0,6922363974 & 0,3269893207 & 0,8844159751 \\ \hline
|
|
5 & 6 & 0,6272474232 & 0,2550197714 & 0,8416304961 \\ \hline
|
|
6 & 7 & 0,5631300932 & 0,1970604773 & 0,7935453363 \\ \hline
|
|
7 & 8 & 0,5024492711 & 0,1513348122 & 0,7435936244 \\ \hline
|
|
8 & 9 & 0,4464575453 & 0,1157397704 & 0,6939114179 \\ \hline
|
|
9 & 10 & 0,3962955817 & 0,0884254966 & 0,646879706 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
On obtient finalement les probabilités recherchées pour calculer les éléments d'assurances
|
|
|
|
\begin{center}\begin{tabular}{|r|r|r|r|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$k$} & \multicolumn{1}{l|}{${}_{k|}q_{xy};\alpha=0,05$} & \multicolumn{1}{l|}{${}_{k|}q_{xy};\alpha=0,2$} & \multicolumn{1}{l|}{${}_{k|}q_{xy};\alpha=0,001$} \\ \hline
|
|
0 & 0,0658166068 & 0,195940902 & 0,0189015891 \\ \hline
|
|
1 & 0,0641257678 & 0,159504556 & 0,0214572292 \\ \hline
|
|
2 & 0,0646300979 & 0,1309912763 & 0,0266710133 \\ \hline
|
|
3 & 0,0636047788 & 0,1064423092 & 0,0305228356 \\ \hline
|
|
4 & 0,0655077884 & 0,08765239 & 0,0383729251 \\ \hline
|
|
5 & 0,0678849598 & 0,0720993881 & 0,0476928265 \\ \hline
|
|
6 & 0,0638832004 & 0,0568116968 & 0,049023241 \\ \hline
|
|
7 & 0,0596832535 & 0,0445193877 & 0,0497904926 \\ \hline
|
|
8 & 0,0540320154 & 0,0343492154 & 0,0479433292 \\ \hline
|
|
9 & 0,0484062949 & 0,0263582742 & 0,0453856021 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
\begin{center}\begin{tabular}{|r|r|r|r|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{$k$} & \multicolumn{1}{l|}{${}_{k|}q_{\overline{xy}};\alpha=0,05$} & \multicolumn{1}{l|}{${}_{k|}q_{\overline{xy}};\alpha=0,2$} & \multicolumn{1}{l|}{${}_{k|}q_{\overline{xy}};\alpha=0,001$} \\ \hline
|
|
0 & 0,0488466739 & 0,1813347454 & 0,0010794202 \\ \hline
|
|
1 & 0,046651046 & 0,1485934848 & 0,0012882257 \\ \hline
|
|
2 & 0,0446980002 & 0,121836356 & 0,0016746589 \\ \hline
|
|
3 & 0,0429409984 & 0,0999313398 & 0,0022215351 \\ \hline
|
|
4 & 0,0416282314 & 0,0821089658 & 0,0032793372 \\ \hline
|
|
5 & 0,0408507573 & 0,0676167361 & 0,0050717995 \\ \hline
|
|
6 & 0,039885391 & 0,0555469896 & 0,0067188655 \\ \hline
|
|
7 & 0,0390101483 & 0,0456019639 & 0,0085836319 \\ \hline
|
|
8 & 0,03789567 & 0,0373244266 & 0,0101818659 \\ \hline
|
|
9 & 0,0366160467 & 0,0304714927 & 0,011562419 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
\begin{center}\begin{tabular}{|r|r|r|r|}
|
|
\hline
|
|
\multicolumn{1}{|l|}{k} & \multicolumn{1}{l|}{$P(T(x)=k,T(y)=k);\alpha=0,05$} & \multicolumn{1}{l|}{$P(T(x)=k,T(y)=k);\alpha=0,2$} & \multicolumn{1}{l|}{$P(T(x)=k,T(y)=k);\alpha=0,001$} \\ \hline
|
|
0 & 0,0488466739 & 0,1813347454 & 0,0010794202 \\ \hline
|
|
1 & 0,045656633 & 0,1458222844 & 0,0010864607 \\ \hline
|
|
2 & 0,0425821835 & 0,1169329054 & 0,0011317243 \\ \hline
|
|
3 & 0,0394742329 & 0,0931991849 & 0,0011673773 \\ \hline
|
|
4 & 0,0364874886 & 0,0739441711 & 0,0012959716 \\ \hline
|
|
5 & 0,0334667679 & 0,0581059955 & 0,0015110506 \\ \hline
|
|
6 & 0,0301891533 & 0,0450208778 & 0,0015426425 \\ \hline
|
|
7 & 0,0271017492 & 0,0347060272 & 0,0015718957 \\ \hline
|
|
8 & 0,0241636367 & 0,0266061291 & 0,0015200819 \\ \hline
|
|
9 & 0,0214852434 & 0,0203513757 & 0,0014419331 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
On peut maintenant calculer la valeur de la PUN
|
|
|
|
\begin{center}\begin{tabular}{|l|r|r|r|}
|
|
\hline
|
|
$\alpha$ & 0,05 & 0,2 & 0,001 \\ \hline
|
|
$\lcterm{A}{xy}{10}$ & 0,4203093983 & 0,6920828305 & 0,2361863918 \\ \hline
|
|
$\lcterm{A}{\overline{xy}}{10}$ & 0,2866104611 & 0,6537938062 & 0,0295011256 \\ \hline
|
|
$A_{(K(x)=K(y)):\lcroof{10}}$ & 0,2474089395 & 0,6089488843 & 0,008699296 \\ \hline
|
|
PUN & 30126,5175707891 & 36034,719133914 & 24656,8186263987 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
On doit maintenant calculer le facteur de rente $\ddot{a}_{\overline{xy}:\lcroof{10}}$
|
|
|
|
\begin{align*}
|
|
\ddot{a}_{\overline{xy}:\lcroof{10}} &= \sum_{k=0}^9 v^k {}_{k}p_{\overline{xy}} \\
|
|
&= \sum_{k=0}^9 v^k ({}_{k}p_{x} + {}_{k}p_{y} - {}_{k}p_{\overline{xy}}) \\
|
|
&= \sum_{k=0}^9 v^k (S_{T(x)}{(k)}+S_{T(y)}{(k)}-S_{T(x)T(y)}{(k,k)})
|
|
\end{align*}
|
|
|
|
\begin{center}\begin{tabular}{|l|r|r|r|}
|
|
\hline
|
|
$\alpha$ & 0,05 & 0,2 & 0,001 \\ \hline
|
|
$\ddot{a}_{\overline{xy}:\lcroof{10}}$ & 5,9978442416 & 3,8631708141 & 7,1716694855 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
On obtient donc les primes annuelles
|
|
|
|
\begin{center}\begin{tabular}{|l|r|r|r|}
|
|
\hline
|
|
\multicolumn{1}{|r|}{$\alpha$} & 0,05 & 0,2 & 0,001 \\ \hline
|
|
$\pi$ & 5022,89095169 & 9327,7571373309 & 3438,0863028259 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
\subsection{Partie d}
|
|
\label{sec:ex3ptd}
|
|
|
|
On utilise la méthode prospective pour $j=2$
|
|
|
|
\begin{align*}
|
|
{}_jV = VPA_{@j}^{PREST} - VPA_{@j}^{PRIMES}
|
|
\end{align*}
|
|
|
|
On évalue d'abord la valeur $VPA_{@j}^{PREST}$ des prestations à verser
|
|
|
|
\begin{align*}
|
|
VPA_{@j}^{PREST} = 100000 \lcterm{A}{xy+2}{8} + 75000 \lcterm{A}{\overline{xy+2}}{8} - 135000 A_{(K(x+2)=K(y+2)):\lcroof{8}}
|
|
\end{align*}
|
|
|
|
\begin{align*}
|
|
\lcterm{A}{xy+2}{8} &= \frac{1}{{}_2E_{xy}}\sum_{k=2}^9 v^{k+1} {}_{k|}q_{xy} \\
|
|
&= \frac{1}{{}_2E_{xy}} \sum_{k=2}^9 v^{k+1} \left[S_{T(x)T(y)}{(k,k)} - S_{T(x)T(y)}{(k+1,k+1)}\right] \\
|
|
\lcterm{A}{\overline{xy+2}}{8} &= \frac{1}{{}_2E_{xy}} \sum_{k=2}^9 v^{k+1} {}_{k|}q_{\overline{xy}} \\
|
|
&= \frac{1}{{}_2E_{xy}} \sum_{k=2}^9 v^{k+1} \bigg\{\left[S_{T(x)}{(k)} + S_{T(y)}{(k)} - S_{T(x)T(y)}{(k,k)} \right] \\
|
|
&\quad - \left[S_{T(x)}{(k+1)} + S_{T(y)}{(k+1)} - S_{T(x)T(y)}{(k+1,k+1)} \right]\bigg\} \\
|
|
\lcterm{A}{(T(x+2)=T(y+2))}{8} &= \frac{1}{{}_2E_{xy}} \sum_{k=2}^9 v^{k+1} P(T(x)=k,T(y)=k) \\
|
|
&= \frac{1}{{}_2E_{xy}} \sum_{k=2}^9 v^{k+1} \left[S_{T(x)T(y)}{(k,k)} - S_{T(x)T(y)}{(k+1,k)} - S_{T(x)T(y)}{(k,k+1)} + S_{T(x)T(y)}{(k+1,k+1)} \right]
|
|
\end{align*}
|
|
|
|
On évalue ensuite la valeur $VPA_{@j}^{PRIMES}$ des primes à recevoir
|
|
|
|
\begin{align*}
|
|
\pi \ddot{a}_{\overline{xy+2}:\lcroof{8}} &= \pi \frac{1}{{}_2E_{xy}} \sum_{k=2}^9 v^k ({}_{k}p_{x} + {}_{k}p_{y} - {}_{k}p_{\overline{xy}}) \\
|
|
&= \pi \frac{1}{{}_2E_{xy}} \sum_{k=2}^9 v^k (S_{T(x)}{(k)}+S_{T(y)}{(k)}-S_{T(x)T(y)}{(k,k)})
|
|
\end{align*}
|
|
|
|
On calcule ${}_2E_{xy} = {}_2p_{xy} v^2$ selon les 3 hypothèses
|
|
|
|
\begin{center}\begin{tabular}{|l|r|r|r|}
|
|
\hline
|
|
$\alpha$ & 0,05 & 0,2 & 0,001 \\ \hline
|
|
${}_2p_{xy}$ & 0,8700576254 & 0,6445545419 & 0,9596411817 \\ \hline
|
|
${}_2E_{xy}$ & 0,7459341782 & 0,5526016306 & 0,8227376386 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
En utilisant les tableaux de la partie c), on obtient les réserves suivantes:
|
|
|
|
\begin{center}\begin{tabular}{|l|r|r|r|}
|
|
\hline
|
|
$\alpha$ & 0,05 & 0,2 & 0,001 \\ \hline
|
|
$\lcterm{A}{xy+2}{8}$ & 0,3043905856 & 0,3739066214 & 0,2002888048 \\ \hline
|
|
$\lcterm{A}{\overline{xy+2}}{8} $ & 0,2013863066 & 0,3584963012 & 0,0273972165 \\ \hline
|
|
$A_{(K(x+2)=K(y+2)):\lcroof{8}}$ & 0,163037334 & 0,316027237 & 0,0067683679 \\ \hline
|
|
$VPA_{@2}^{PREST}$ & 31548,3485751192 & 39113,5431854715 & 25731,0970754847 \\ \hline
|
|
$\ddot{a}_{\overline{xy+2}:\lcroof{8}}$ & 4,1171467174 & 2,1051474302 & 5,2467430227 \\ \hline
|
|
$VPA_{@2}^{PRIMES}$ & 27723,5976031596 & 35534,2852419522 & 21925,2827068092 \\ \hline
|
|
${}_2V$ & 3824,7509719595 & 3579,2579435192 & 3805,8143686755 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
\subsection{Partie e}
|
|
\label{sec:ex3pte}
|
|
|
|
Comme il ne reste qu'un seul assuré vivant, la seule prestation qui pourra être versée est celle de l'assurance dernier décès. La prime continue cependant d'être payée.
|
|
|
|
${}_2V = VPA_{@2}^{PREST} - VPA_{@2}^{PRIMES}$
|
|
|
|
En utilisant la méthode prospective et en considérant la situation où l'homme est survivant, on obtient, pour la partie des prestations
|
|
\begin{align*}
|
|
VPA_{@2}^{PREST} &= 75000 \lcterm{A}{x+2}{8} \\
|
|
&= 75000 \frac{1}{{}_2E_x} \sum_{k=2}^{9} v^{k+1} {}_kp_x q_{x+k} \\
|
|
\end{align*}
|
|
et pour la partie des primes
|
|
\begin{align*}
|
|
VPA_{@2}^{PRIMES} &= \pi \ddot{a}_{x+2:\lcroof{8}} \\
|
|
&= \pi \frac{1}{{}_2E_x} \sum_{i=1}^n v^{k} {}_kp_x
|
|
\end{align*}
|
|
|
|
En considérant la situation ou la femme est survivante, on obtient, pour la partie des prestations
|
|
\begin{align*}
|
|
VPA_{@2}^{PREST} &= 75000 \lcterm{A}{y+2}{8} \\
|
|
&= 75000 \frac{1}{{}_2E_y} \sum_{k=2}^{9} v^{k+1} {}_kp_y q_{y+k} \\
|
|
\end{align*}
|
|
et pour la partie des primes
|
|
\begin{align*}
|
|
VPA_{@2}^{PRIMES} &= \pi \ddot{a}_{y+2:\lcroof{8}} \\
|
|
&= \pi \frac{1}{{}_2E_y} \sum_{i=1}^n v^{k} {}_kp_y
|
|
\end{align*}
|
|
|
|
On obtient alors comme résultat:
|
|
|
|
Homme survivant
|
|
|
|
\begin{center}\begin{tabular}{|l|r|r|r|}
|
|
\hline
|
|
$\alpha$ & 0,05 & 0,2 & 0,001 \\ \hline
|
|
$\pi$ & 5022,89095169 & 9327,7571373309 & 3438,0863028259 \\ \hline
|
|
$VPA_{@2}^{PREST}$ & 10560,3958489596 & 10560,3958489596 & 10560,3958489596 \\ \hline
|
|
$VPA_{@2}^{PRIMES}$ & 29078,83283987 & 54000,8320260448 & 19903,9831544191 \\ \hline
|
|
${}_2V$ & -18518,4369909104 & -43440,4361770852 & -9343,5873054595 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
Femme survivante
|
|
|
|
\begin{center}\begin{tabular}{|l|r|r|r|}
|
|
\hline
|
|
$\alpha$ & 0,05 & 0,2 & 0,001 \\ \hline
|
|
$\pi$ & 5022,89095169 & 9327,7571373309 & 3438,0863028259 \\ \hline
|
|
$VPA_{@2}^{PREST}$ & 10421,4909345379 & 10421,4909345379 & 10421,4909345379 \\ \hline
|
|
$VPA_{@2}^{PRIMES}$ & 29441,303579022 & 54673,9581314525 & 20152,0884179894 \\ \hline
|
|
${}_2V$ & -19019,8126444841 & -44252,4671969146 & -9730,5974834515 \\ \hline
|
|
\end{tabular}\end{center}
|
|
|
|
|
|
|
|
\clearpage
|
|
|
|
\includegraphics[height=7mm,keepaspectratio=true]{by-sa}\\%
|
|
Cette création est mise à disposition selon le contrat
|
|
\href{http://creativecommons.org/licenses/by-sa/2.5/ca/deed.fr}{%
|
|
Paternité-Partage à l'identique 2.5 Canada} de Creative Commons
|
|
disponible à l'adresse
|
|
http://creativecommons.org/licenses/by-sa/2.5/ca/deed.fr. En vertu de ce
|
|
contrat, vous êtes libre de:
|
|
\begin{itemize}
|
|
\item \textbf{partager} --- reproduire, distribuer et communiquer
|
|
l'{\oe}uvre;
|
|
\item \textbf{remixer} --- adapter l'{\oe}uvre;
|
|
\item utiliser cette {\oe}uvre à des fins commerciales.
|
|
\end{itemize}
|
|
Selon les conditions suivantes:
|
|
|
|
\begin{tabularx}{\linewidth}{@{}lX@{}}
|
|
\raisebox{-9mm}[0mm][13mm]{%
|
|
\includegraphics[height=11mm,keepaspectratio=true]{by}} &
|
|
\textbf{Attribution} --- Vous devez attribuer l'{\oe}uvre de la
|
|
manière indiquée par l'auteur de l'{\oe}uvre ou le titulaire des
|
|
droits (mais pas d'une manière qui suggérerait qu'ils vous
|
|
soutiennent ou
|
|
approuvent votre utilisation de l'{\oe}uvre). \\
|
|
\raisebox{-9mm}{\includegraphics[height=11mm,keepaspectratio=true]{sa}}
|
|
& \textbf{Partage à l'identique} --- Si vous modifiez, transformez
|
|
ou adaptez cette {\oe}uvre, vous n'avez le droit de distribuer
|
|
votre création que sous une licence identique ou similaire à
|
|
celle-ci.
|
|
\end{tabularx}
|
|
\end{document}
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|
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