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-14,7 +15,11 @@ \usepackage{array} \newcolumntype{C}[1]{>{\centering\arraybackslash$}p{#1}<{$}} \usepackage{rotating} +\usepackage{pdflscape} +\usepackage{hyperref} +\usepackage{mathpazo}% Palatino mit mathematischen Zeichen +\hyphenation{Prä-mi-en-frei-stel-lung} %opening \title{\textbf{Formelsammlung Tarif Lebensversicherung}} @@ -29,6 +34,7 @@ \newcommand{\orZero}[1]{\mathit{\left[#1\right]}} \newcommand{\xn}{{\act[x]{n}}} + \setlength{\parindent}{0em} \makeatletter @@ -61,62 +67,173 @@ Bemerkung: Sämtliche Barwerte werden rekursiv bestimmt, daher werden alle Formeln in ihrer rekursiven Form angegeben. Teilweise wird aus informativen Gründen davor die entsprechende Summenformel angegeben, diese wird jedoch nicht zur tatsächlichen Berechnung benutzt. +\tableofcontents + + +\begin{landscape} +\section{Definitionen sämtlicher Variablen} + +\subsection{Vertragsdetails (vertragsspezifische Werte)} + +\newenvironment{deftab}{\begin{longtable}{p{1.7cm}p{13cm}p{9cm}}}{\end{longtable}} + +\begin{deftab} + $VS$ & Versicherungssumme & \texttt{contract\$params\$sumInsured}\\ + $\widetilde{VS}$ & Versicherungssumme nach außerplanmäßiger Prä"-mi"-en"-frei"-stel"-lung\\[0.5em] + $x$ & Eintrittsalter der versicherten Person & \texttt{contract\$params\$age}\\ + $n$ & Versicherungsdauer & \texttt{contract\$params\$policyPeriod}\\ + $l$ & Aufschubdauer des Versicherungsschutzes & \texttt{contract\$params\$deferral}\\ + $m$ & Prämienzahlungsdauer & \texttt{contract\$params\$premiumPeriod}\\ + $k$ & Prämienzahlungsweise ($k$-tel jährlich) & \texttt{contract\$params\$premiumFrequency}\\ + $g$ & Garantiedauer (für Renten) & \texttt{contract\$params\$guaranteed}\\ + $f$ & Prämienfreistellungszeitpunkt\\[0.5em] + + $YOB$ & Geburtsjahr der versicherten Person (bei Benutzung von Generationentafeln) & \texttt{contract\$params\$YOB}\\ + $Beg$ & Versicherungsbeginn (Datum, TODO)\\[0.5em] + + $q_{x+t}$ & einjährige Sterbewahrscheinlichkeit der versicherten Person (aus $YOB$ und $x$ bestimmt) & \texttt{contract\$params\$transitionProbabilities\$q}\\ + $p_{x+t}$ & einjährige Überlebenswahrscheinlichkeit, $p_{x+t} = 1-q_{x+t}$ & \texttt{contract\$params\$transitionProbabilities\$p} \\ + $\omega$ & Höchstalter gemäß der benutzten Sterbetafel & \texttt{getOmega(tarif\$mortalityTable)} \\[0.5em] + -\section{Vertrags- und Tarifdetails} + $k_{Ausz}$ & unterjährige Auszahlung der Erlebensleistungen (nur Renten) & \texttt{contract\$params\$benefitFrequency}\\ + $y$ & Eintrittsalter der 2. versicherten Person (TODO)\\ + + +\end{deftab} \subsection{Tarifdetails (identisch für alle Verträge)} -\begin{longtable}{p{1cm}p{10cm}} - $i$ & Rechnungszins\\[0.5em] - $v$ & Diskontierungsfaktor $v=\frac1{1+i}$\\[0.5em] +\begin{deftab} + $i$ & Rechnungszins & \texttt{tarif\$i}\\[0.5em] + $v$ & Diskontierungsfaktor $v=\frac1{1+i}$ & \texttt{tarif\$v}\\[0.5em] - ${}$ & Leistungszahlungsweise (vorschüssig, nachschüssig)\\ - $k_{Ausz}$ & unterjährige Auszahlung der Erlebensleistungen\\ - $O(k)$ & Ordnung der Unterjährigkeitsrechnung (1./2. Ordnung)\\[0.5em] + $\rho$ & Sicherheitszuschlag auf die Prämie & \texttt{tarif\$loadings\$security} \\ + $\rho^{RG}$ & Risikosumme (relativ zu DK) im Ablebensfall bei Prämienrückgewähr & \texttt{tarif\$premiumRefundLoading} \\ + $uz(k)$ & Unterjährigkeitszuschlag bei $k$-tel jährlicher Prämienzahlung (in \% der Prämie) & \texttt{tarif\$premiumFrequencyLoading}\\ + $O(k)$ & Ordnung der Unterjährigkeitsrechnung der Erlebenszahlungen (0./""1./""1,5./""2. Ordnung) & \texttt{tarif\$benefitFrequencyOrder}\\[0.5em] - $RG$ & Prämienrückgewähr bei Ableben während Aufschub\\ - $PS$ & Prämiensumme \\ +\end{deftab} + +\subsection{Leistungen} + +\begin{deftab} + $\Pi_t$ & (Netto-)Prämie zum Zeitpunkt $t$ (vorschüssig), $\Pi^{nachsch.}_t$ für nachschüssige Prämienzahlungsweise, normiert auf 1 & \texttt{contract\$cashFlows\$premiums\_advance}\\ + $\Pi^{nachsch.}_t$ & (Netto-)Prämie zum Zeitpunkt $t$ (nachschüssig), normiert auf 1 & \texttt{contract\$cashFlows\$premiums\_arrears}\\ + $PS_t$ &Bruttoprämiensumme bis zum Zeitpunkt $t$: $PS_t=\sum_{j=0}^t \Pi^a_t$\\ + $PS$ & Bruttoprämiensumme über die gesamte Vertragslaufzeit & \texttt{contract\$premiumSum}\\[0.5em] - $\rho$ & Sicherheitszuschlag auf die Prämie \\ - $\rho^{RG}$ & Risikosumme (relativ zu DK) im Ablebensfall bei Prämienrückgewähr \\ + $\ddot{e}_t$ & vorschüssige Erlebenszahlung zum Zeitpunkt $t$ (normiert auf 1) & \texttt{contract\$cashFlows\$survival\_advance}\\ + $e_t$ & nachschüssige Erlebenszahlung zum Zeitpunkt $t$ (normiert auf 1) & \texttt{contract\$cashFlows\$survival\_arrears}\\[0.5em] -\end{longtable} + $\ddot{e}_t^*$ & vorschüssige garantierte Zahlung zum Zeitpunkt $t$ (normiert auf 1) & \texttt{contract\$cashFlows\$guaranteed\_advance}\\ + $e_t^*$ & nachschüssige garantierte Zahlung zum Zeitpunkt $t$ (normiert auf 1) & \texttt{contract\$cashFlows\$guaranteed\_arrears}\\[0.5em] + + $a_t$ & Ablebensleistung proportional zur Versicherungssumme (nachschüssig) & \texttt{contract\$cashFlows\$death\_sumInsured}\\ + $a_t^{(RG)}$ & Ablebensleistung für Prämienrückgewähr (normiert auf Prämie 1, relativ zu Prämienzumme $PS$) & \texttt{contract\$cashFlows\$death\_GrossPremium}\\[0.5em] + $\widetilde{a}_t$ & Ablebensleistung nach außerplanmäßiger Prämienfreistellung proportional zur Versicherungssumme (nachschüssig) & \texttt{contract\$cashFlows\$death\_PremiumFree}\\ + + $\overrightarrow{CF}^B_t$ & Leistungscashflow (relativ zur jeweiligen Basis, sowie vor"~/""nachschüssig) als Matrix dargestellt +\end{deftab} +\subsection{Kosten} +Mögliche Basen für die Kostenfestsetzung sind: +\begin{align*} + \text{Basis} &= \begin{cases}\text{VS ... Versicherungssumme (\texttt{"{}SumInsured"})}\\\text{PS ... gesamte Prämiensumme (\texttt{"{}SumPremiums"})}\\\text{BP ... Bruttojahresprämie (\texttt{"{}GrossPremium"})}\end{cases} & + \text{Dauer} &= \begin{cases}\text{1 ... einmalig bei Abschluss (\texttt{"{}once"})}\\\text{PD ... Prämienzahlungsdauer (\texttt{"{}PremiumPeriod"})}\\\text{Prf ... Nach Ablauf der Prämienzahlungsdauer (\texttt{"{}PremiumFree"})}\\\text{LZ ... gesamte Laufzeit (\texttt{"{}PolicyPeriod"})}\end{cases} +\end{align*} -\subsection{Vertragsdetails (vertragsspezifische Werte)} -\begin{longtable}{p{1cm}p{10cm}} - $VS$ & Versicherungssumme\\[0.5em] - $Beg$ & Versicherungsbeginn\\ +\begin{deftab} + $\alpha_t^{\text{Basis},\text{Dauer}}$ & Abschlusskostensatz relativ zu Basis über die angegebene Dauer & \texttt{tarif\$costs["{}alpha",,]}\\ + $Z_t^{\text{Basis},\text{Dauer}}$ & Zillmerkostensatz relativ zu Basis über die angegebene Dauer & \texttt{tarif\$costs["{}Zillmer",,]}\\ + $\beta_t^{\text{Basis},\text{Dauer}}$ & Inkassokostensatz relativ zu Basis über die angegebene Dauer & \texttt{tarif\$costs["{}beta",,]}\\ + $\gamma_t^{\text{Basis},\text{Dauer}}$ & Verwaltungskostensatz relativ zu Basis über die angegebene Dauer & \texttt{tarif\$costs["{}gamma",,]}\\ + $\widetilde{\gamma}_t^{\text{Basis},\text{Dauer}}$ & Verwaltungskostensatz nach außerplanmäßiger Prämienfreistellung relativ zu Basis über die angegebene Dauer & \texttt{tarif\$costs["{}gamma\_nopremiums",,]}\\[0.5em] - $J$ & Geburtsjahr der 1. versicherten Person\\ - $x$ & Eintrittsalter der 1. versicherten Person\\ - $y$ & Eintrittsalter der 2. versicherten Pereon\\[0.5em] + $\alpha_t^{\text{Basis}}$ & Abschlusskosten-Cash Flow relativ zu Basis zu $t$ & \texttt{contract\$cashFlowsCosts[,"{}alpha", Basis]}\\ + $Z_t^{\text{Basis}}$ & Zillmerkosten-Cash Flow relativ zu Basis zu $t$ & \texttt{contract\$cashFlowsCosts[,"{}Zillmer", Basis]}\\ + $\beta_t^{\text{Basis}}$ & Inkassokosten-Cash Flow relativ zu Basis zu $t$ & \texttt{contract\$cashFlowsCosts[,"{}beta", Basis]}\\ + $\gamma_t^{\text{Basis}}$ & Verwaltungskosten-Cash Flow relativ zu Basis zu $t$ & \texttt{contract\$cashFlowsCosts[,"{}gamma", Basis]}\\ + $\widetilde{\gamma}_t^{\text{Basis}}$ & Verwaltungskosten-Cash Flow nach außerplanmäßiger Prämienfreistellung relativ zu Basis zu $t$ & \texttt{contract\$cashFlowsCosts[,"{}gamma\_nopremiums", Basis]}\\[0.5em] + + $\overrightarrow{CF}^C_t$ & Kostencashflows (relativ zur jeweiligen Basis) als Matrix dargestellt & \texttt{contract\$cashFlowsCosts["{}t",,]} +\end{deftab} + +\subsection{Barwerte} + +\begin{deftab} + $P_\xn(t)$ & BW der zuk. Prämienzahlungen (mit Prämie 1) zum Zeitpunkt $t$ & \texttt{contract\$presentValues\$premiums}\\ + $E^*_\xn(t)$ & BW der zuk. garantierten Zahlungen (mit VS 1) zum Zeitpunkt $t$ & \texttt{contract\$presentValues\$guaranteed}\\ + $E_\xn(t)$ & BW der zuk. Erlebenszahlungen (mit VS 1) zum Zeitpunkt $t$ & \texttt{contract\$presentValues\$survival}\\ + $E^{(k)}_\xn(t)$ & BW der zuk. Erlebenszahlungen (mit VS 1) bei $k$-tel jährlicher Auszahlung zum Zeitpunkt $t$ & \texttt{contract\$presentValues\$survival}\\ + $\alpha(k)$, $\beta(k)$ & Unterjährigkeitskorrektur bei $k$-tel jährlicher Auszahlung\\ + $A_\xn(t)$ & BW der zuk. Ablebensleistungen (mit VS 1) zum Zeitpunkt $t$ & \texttt{contract\$presentValues\$death\_SumInsured}\\ + $A_\xn^{prf}(t)$ & BW der zuk. Ablebensleistungen (mit VS 1) zum Zeitpunkt $t$ nach Prämienfreistellung & \texttt{contract\$presentValues\$death\_GrossPremium}\\ + $A^{(RG)}_\xn(t)$ & BW der zuk. Ablebensleistungen aus Prämienrückgewähr (mit BP 1) zum Zeitpunkt $t$ & \texttt{contract\$presentValues\$death\_PremiumFree}\\[0.5em] - $n$ & Versicherungsdauer\\ - $l$ & Aufschubdauer des Versicherungsschutzes\\ - $m$ & Prämienzahlungsdauer\\ - $k$ & Prämienzahlungsweise ($k$-tel jährlich)\\ - $f$ & Prämienfreistellungszeitpunkt\\[0.5em] + $PV^{B1}_\xn(t)$ & BW aller zuk. Leistungen (ohne Prämienrückgewähr) zum Zeitpunkt $t$ & \texttt{contract\$presentValues\$benefits}\\[1em] + $PV^B_\xn(t)$ & BW aller zuk. Leistungen (inkl. Prämienrückgewähr) zum Zeitpunkt $t$ & \texttt{contract\$presentValues\$benefitsAndRefund}\\[1em] -\end{longtable} +\end{deftab} + KOSTEN (TODO) + +\subsection{Prämien und Prämienzerlegung} + +\begin{deftab} + $\Pi^1_\xn$ & Nettoprämie auf VS 1 & \texttt{contract\$premiums[["{}unit.net"]]}\\ + $\Pi_\xn^{1,Z}$ & Zillmerprämie auf VS 1 & \texttt{contract\$premiums[["{}unit.Zillmer"]]}\\ + $\Pi_\xn^{1,a}$ & Bruttoprämie (``adequate'' bzw. ``expense-load premium'') auf VS 1 & \texttt{contract\$premiums[["{}unit.gross"]]}\\[0.5em] + $\Pi_\xn$ & Nettoprämie & \texttt{contract\$premiums[["{}net"]]}\\ + $\Pi_\xn^Z$ & Zillmerprämie & \texttt{contract\$premiums[["{}Zillmer"]]}\\ + $\Pi_\xn^a$ & Bruttoprämie (``adequate'' bzw. ``expense-load premium'') & \texttt{contract\$premiums[["{}gross"]]}\\[0.5em] + + $\Pi_\xn^\alpha$ & $\alpha$-Kostenprämie (Abschlusskostenprämie)\\ + $\Pi_\xn^\beta$ & $\beta$-Kostenprämie (Inkassokostenprämie)\\ + $\Pi_\xn^\gamma$ & $\gamma$-Kostenprämie (Verwaltungskostenprämie)\\[0.5em] + + $\Pi_\xn^\text{inv.}$ & Inventarprämie (Netto- plus Verwaltungskostenprämie)\\[1em] + + $\Pi_\xn^s$ & Sparprämie (zum Aufbau des Nettodeckungskapitals investierter Teil der Prämie)\\ + $\Pi_\xn^r$ & Risikoprämie (zur Deckung des einjährigen Ablebensrisikos benutzter Teil der Prämie)\\[0.5em] + + $\Pi_\xn^v$ & verrechnete Prämie (Bruttoprämie inkl. Rabatte, Zuschläge, Stück"-kosten und Steuer) & \texttt{contract\$premiums[["{}written"]]}\\ + $\Pi_\xn^\text{tax}$ & Versicherungssteuer & \texttt{contract\$premiums[["{}tax"]]}\\ +\end{deftab} -\section{Rechnungsgrundlagen} +\subsection{Absolutwerte der Cashflows und Barwerte} -\begin{longtable}{p{0.2\linewidth}p{0.7\linewidth}} - $q_x=q_x^{(J)}(t)$ & Sterbewahrscheinlichkeit der 1. versicherten Person (geboren im Jahr $J$) im Beobachtungsjahr $t$\\ - $p_x=1-q_x$ & 1-jährige Überlebenswahrscheinlichkeit der 1. versicherten Person\\ - ${}_np_x=\prod_{j=1}^n p_{x+j}$ & $n$-jährige Überlebenswahrscheinlichkeit\\ - $\omega$ & Höchstalter\\ +TODO + +\subsection{Rückstellungen} + +\begin{deftab} + $_{t}V_\xn$ & Nettodeckungskapital zum Zeitpunkt $t$ \\ + $_{t}V^a_\xn$ & Brutto-Deckungskapital (``adequate'' bzw. ``expense-loaded reserve'') \\ + $_{t}V^Z_\xn$ & Zillmerreserve bei Zillmerung \\ + $_{t}V^\alpha_xn$ & Abschlusskostenreserve \\ + $_{t}V^\beta_xn$ & Inkassokostenreserve (typischerweise $=0$)\\ + $_{t}V^\gamma_xn$ & Verwaltungskostenreserve \\ + $_{t}V^{Umr}_\xn$ & Umrechnungsreserve (Basis für Vertragskonvertierungen und Prämienfreistellung), inkl. anteiliger Abschlusskostenrückerstattung bei Beendigung innerhalb von fünf Jahren\\ + $AbskErh(t)$ & Anteilsmäßige Rückerstattung der Abschlusskosten bei Kündigung innerhalb von fünf Jahren\\[1em] - $q_y=q_y^{(J+x-y)}(t)$ & Sterbewahrscheinlichkeit der 2. versicherten Person (geboren im Jahr $J$) im Beobachtungsjahr $t$\\ -\end{longtable} + $BilRes_{t+u}$ & Bilanzreserve (Interpolation aus $_{t}V^{x}_\xn$ und $_{t+1}V^{x}_\xn$) +\end{deftab} + +\subsection{Werte nach außerplanmäßiger Prämienfreistellung} + +Werte nach außerplanmäßiger Prämienfreistellung werden durch ein $\widetilde{\quad}$ über dem jeweiligen Symbol angezeigt. + + + + +\end{landscape} \pagebreak \section{Kosten} -\subsection{Abschlusskosten ($\alpha$-Kosten) / Zillmerkosten (Z-Kosten)} +\subsection{Abschlusskosten (\texorpdfstring{$\alpha$}{alpha}-Kosten) / Zillmerkosten (Z-Kosten)} \begin{itemize} \item Einmalig (bei Vertragsabschluss) \begin{itemize} @@ -140,14 +257,14 @@ Bemerkung: Sämtliche Barwerte werden rekursiv bestimmt, daher werden alle Forme % TODO: Check the following tariffs: % - GW PV5F & PV6F -\subsection{Inkassokosten ($\beta$-Kosten)} +\subsection{Inkassokosten (\texorpdfstring{$\beta$}{beta}-Kosten)} \begin{itemize} \item Laufend an Bruttoprämie während Prämienzahlungsdauer (einmalig bei Einmalerlag) \end{itemize} -\subsection{Verwaltungskosten ($\gamma$-Kosten)} +\subsection{Verwaltungskosten (\texorpdfstring{$\gamma$}{gamma}-Kosten)} Laufend während der gesamten Laufzeit verrechnet: @@ -167,7 +284,7 @@ Laufend während der gesamten Laufzeit verrechnet: % - GW PS0-9 & PV0-9 % - ER III B-PZV Single -\subsection{Stückkosten $StkK$} +\subsection{Stückkosten \textit{StkK}} \begin{itemize} \item Stückkosten (Absolutbetrag) $StkK$ pro Jahr während Prämienzahlungsdauer (bzw. einmalig bei Einmalprämie) @@ -254,7 +371,7 @@ $PS$ & Prämiensumme, $PS=\sum_{t=0}^n pr_t$\\[1em] $\ddot{e}_t$ \dots & Erlebenszahlungen vorschüssig zu $t$ & $\delta(l+g\le t< n)$ & 0 & $\delta(t=n)$\\ $e_t$ \dots & Erlebenszahlungen nachschüssig zu $t+1$& $\delta(l+g\le t< n)$ & 0 & $\delta(t=n)$\\ $\ddot{e}_t^{*}$ \dots & garantierte Zahlungen vorschüssig zu $t$ & $\delta(l\le t< l+g)$ & 0 & 0 \\ -$e_t^{*}$ \dots & garantierte Zahlungen nachschüssig zu $t+1$ & $\delta(l\le t< l+g)$ & 0 & 0 \\[1em] +$e_t^{*}$ \dots & garant. Zahlungen nachschüssig zu $t+1$ & $\delta(l\le t< l+g)$ & 0 & 0 \\[1em] $a_t$ \dots & Ablebenszahlung zu $t+1$ & 0 & $\delta(l\le t < n)$ & 0 \\ $a_t^{(RG)}$ & Ablebenszahlungen für PRG zu $t+1$ (Ableben im Jahr $t$) & $\min(t+1,m,f)$ & 0 & $\min(t+1,m,f)$ \\ @@ -354,7 +471,7 @@ $\beta(m)$ = & $\frac{m-1}{2m}$ & $+ \frac{m^2-1}{6 m^2}i$ & $\left[+ \frac{1-m^ ergibt sich auch für allgemeine unterjährige Erlebenszahlungen $\ddot{e}_t$ eine Rekursionsgleichung. -\subsubsection{Vorschüssige $m$-tel jährliche Auszahlung der Erlebensleistungen} +\subsubsection{Vorschüssige \textit{m}-tel jährliche Auszahlung der Erlebensleistungen} \begin{align*} @@ -363,14 +480,14 @@ A^{(m)}_\xn(t) &= \ddot{e}_t \cdot \ddot{a}_{\act[x+t]{1}}^{(m)} + v \cdot p_{x+ \end{align*} -\subsubsection{Nachschüssige $m$-tel jährliche Auszahlung der Erlebensleistungen} +\subsubsection{Nachschüssige \textit{m}-tel jährliche Auszahlung der Erlebensleistungen} \begin{align*} A^{(m)}_\xn(t) &= e_t \cdot a_{\act[x+t]{1}}^{(m)} + v \cdot p_{x+t} \cdot A^{(m)}_\xn(t+1)\\ &= e_t\cdot\left\{\alpha(m) - \left(\beta(m)+\frac1m\right)\cdot\left(1-p_{x+t} v\right)\right\} + v\cdot p_{x+t} \cdot A^{(m)}_\xn(t+1) \end{align*} -\subsubsection{Allgemeine $m$-tel jährliche Auszahlung der Erlebensleistungen} +\subsubsection{Allgemeine \textit{m}-tel jährliche Auszahlung der Erlebensleistungen} \begin{align*} @@ -518,7 +635,7 @@ BP_\xn &= \frac% \end{align*} Wie man deutlich sehen kann, ist die Kostenursache ($\alpha$, $\beta$ oder $\gamma$) für die Prämienbestimmung irrelevant. Es werden die Barwerte aller drei Kostenarten jeweils bei der entsprechenden Bemessungsgrundlage aufaddiert. % -\subsection{Ablebensleistung im Jahr $t$:} +\subsection{Ablebensleistung im Jahr \textit{t}:} \begin{align*} Abl(t) &= \left\{a_t + a^{(RG)}_t \cdot BP_\xn\right\} \cdot VS \end{align*} -- GitLab