From 20d5854b9322ed091660b0cb0936878d405b8503 Mon Sep 17 00:00:00 2001 From: Reinhold Kainhofer <reinhold@kainhofer.com> Date: Sun, 24 Apr 2016 22:09:32 +0200 Subject: [PATCH] Update some of the formulas up to the surrender value (lots of work still to be done) --- .../Formelsammlung_Beispielrechnung.pdf | Bin 221399 -> 258738 bytes .../Formelsammlung_Beispielrechnung.tex | 418 +++++++++++++----- 2 files changed, 306 insertions(+), 112 deletions(-) diff --git a/Formulas_Reference/Formelsammlung_Beispielrechnung.pdf b/Formulas_Reference/Formelsammlung_Beispielrechnung.pdf index 105f4b4c6cd5eacdebfe1a8419048aca6ca26b3f..5513630f8e86d7ebac687efecdec3388b3d989ba 100644 GIT binary patch delta 179606 zcmcb9kayEl{t2p#CX*k!X-@Qi#HhO2h_QmP-cPVOz;NBA@DqFtZ`Q?ppP9^YqtX4$ z#x-r5@_sdcU^Wk!z28zePDheezBb2I$@I#OgANwXcdZ{4t`FnWy2Wr)L_$6EN&f<+ z9*?HyX2(7(^_JUnt2%kV!GS(EH<v)C?#?AevAtVZ_%=;Qo56DL=>&~E3T!V<ty|y! z)1#?!ietTvjlJLY)%6Y;w`ATWNBy}jd3nhR+co#jYyMpK&fLs0M()bA1HVIeO1|6O 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\newcolumntype{v}[1]{>{\PreserveBackslash\centering\hspace{0pt}}p{#1}} \newcommand{\markiert}[1]{\colorbox{Goldenrod}{\textcolor{red}{#1}}} +\newcommand{\orZero}[1]{\mathit{\left[#1\right]}} +\newcommand{\xn}{{\act[x]{n}}} \setlength{\parindent}{0em} @@ -120,7 +122,7 @@ Bemerkung: Sämtliche Barwerte werden rekursiv bestimmt, daher werden alle Forme \begin{itemize} \item \markiert{an Versicherungssumme} \item \markiert{an Brutto-Prämiensumme}\footnote{Entspricht Einmalprämie bei Einmalerlag} - \item an Barwert der Versicherungsleistungen (z.B. Rentenbarwert) % HYPO PV5F & PV6F + \item an Barwert der Versicherungsleistungen (z.B. Rentenbarwert) % HP PV5F & PV6F \end{itemize} \item Laufend (während Prämienzahlungsdauer)\footnote{Bei Einmalerlag sind einmalige $\alpha$-Kosten und laufende $\alpha$-Kosten auf die Prämie während der Prämienzahlungsdauer ident.} \begin{itemize} @@ -136,7 +138,7 @@ Bemerkung: Sämtliche Barwerte werden rekursiv bestimmt, daher werden alle Forme \end{itemize} % TODO: Check the following tariffs: -% - GRAWE PV5F & PV6F +% - GW PV5F & PV6F \subsection{Inkassokosten ($\beta$-Kosten)} @@ -152,18 +154,18 @@ Laufend während der gesamten Laufzeit verrechnet: \begin{itemize} \item \markiert{an Versicherungssumme (prämienpflichtig)} \item \markiert{an Versicherungssumme (planmäßig/außerplanmäßig prämienfrei)} - \item an Leistungsbarwert / Rentenbarwert (=Deckungskapital) (prämienfrei) % GRAWE PS0-9 & PV0-9 + \item an Leistungsbarwert / Rentenbarwert (=Deckungskapital) (prämienfrei) % GW PS0-9 & PV0-9 \item an Prämiensumme (prämienpflichtig) (=am Rentenbarwert zu Vertragsbeginn bei sof.beg.LR mit EE) \item an Prämiensumme (planmäßig/außerplanmäßig prämienfrei) - \item am Ablösekapital während Aufschubzeit % ERGO III B-PZV Single - \item \markiert{an jeder Erlebenszahlung/Rente (während Liquiditätsphase)} % ERGO III B-PZV Single - \item am Deckungskapital % S-Versicherung I5U + \item am Ablösekapital während Aufschubzeit % ER III B-PZV Single + \item \markiert{an jeder Erlebenszahlung/Rente (während Liquiditätsphase)} % ER III B-PZV Single + \item am Deckungskapital % SV I5U \end{itemize} % TODO: Check the following tariffs: -% - GRAWE PS0-9 & PV0-9 -% - ERGO III B-PZV Single +% - GW PS0-9 & PV0-9 +% - ER III B-PZV Single \subsection{Stückkosten $StkK$} @@ -289,19 +291,19 @@ z^{(VS)}_t & z^{(PS)}_t & -\\ \subsection{Prämienbarwert} \begin{align*} -P_{\act[x]{n}}(t) &= \sum_{j=t}^n pr_{t+j} \cdot v^{j-t} \cdot {}_{j-t}p_{x+t}\\ - &= pr_{t} + v \cdot p_{x+t} \cdot P_{\act[x]{n}}(t+1) +P_\xn(t) &= \sum_{j=t}^n pr_{t+j} \cdot v^{j-t} \cdot {}_{j-t}p_{x+t}\\ + &= pr_{t} + v \cdot p_{x+t} \cdot P_\xn(t+1) \end{align*} \subsection{Barwert garantierter Zahlungen:} Garantierte Erlebensleistungen (wenn Aufschubzeit überlebt wurde): % TODO! \begin{align*} - E^{Gar}_{\act[x]{n}}(t) &= \begin{cases} + E^{Gar}_\xn(t) &= \begin{cases} {}_{l-t}p_{x+t} \cdot v^{l-t} \cdot \sum_{j=l}^{n} \left\{\ddot{e}^{*}_{j-t}+ v\cdot e^*_{j-t}\right\} v^{j-t} & \text{für $t<l$ (Aufschubzeit)}\\ \sum_{j=t}^{n} \left\{\ddot{e}^{*}_{j-t}+ v\cdot e^*_{j-t}\right\} v^{j-t} & \text{für $t\ge l$ (Liquiditätsphase)} \end{cases}\\ - &= \ddot{e}_t^{*} + \left\{E^{*}_{\act[x]{n}}(t+1) + e_t^*\right\}\cdot v \cdot \begin{cases} + &= \ddot{e}_t^{*} + \left\{E^{*}_\xn(t+1) + e_t^*\right\}\cdot v \cdot \begin{cases} 1 & \text{für $t<l$ (Aufschubzeit)}\\ p_{x+t} & \text{für $t\ge l$ (Liquiditätsphase)} \end{cases} @@ -311,8 +313,8 @@ Garantierte Erlebensleistungen (wenn Aufschubzeit überlebt wurde): \subsection{Erlebensleistungsbarwert:} 1. Person: \begin{align*} -E_{\act[x]{n}}(t) &= \sum_{j=t}^n \left(\ddot{e}_{t+j} \cdot v^{j-t} {}_{j-t}p_{x+t} + e_{t+j} \cdot v^{j+1-t} {}_{j+1-t}p_{x+t} \right)\\ - &= \ddot{e}_{t} + v \cdot p_{x+t} \cdot \left\{ e_t + E_{\act[x]{n}}(t+1)\right\}\\ +E_\xn(t) &= \sum_{j=t}^n \left(\ddot{e}_{t+j} \cdot v^{j-t} {}_{j-t}p_{x+t} + e_{t+j} \cdot v^{j+1-t} {}_{j+1-t}p_{x+t} \right)\\ + &= \ddot{e}_{t} + v \cdot p_{x+t} \cdot \left\{ e_t + E_\xn(t+1)\right\}\\ \intertext{2. Person:} E2_{\act[y]{n}}(t) &= \ddot{e}_{t} + v \cdot p_{y+t} \cdot \left\{ e_t + E2_{\act[y]{n}}(t+1)\right\}\\ \intertext{gemeinsam:} @@ -321,11 +323,11 @@ E12_{\act[x,y]{n}}(t) &= \ddot{e}_{t} + v \cdot p_{x+t} \cdot p_{y+t} \cdot \lef % \intertext{Garantierte Erlebensleistungen (wenn Aufschubzeit überlebt wurde):} % % \begin{align*} -% E^{*}_{\act[x]{n}}(t) &= \begin{cases} +% E^{*}_\xn(t) &= \begin{cases} % {}_{l-t}p_{x+t} \cdot v^{l-t} \cdot \sum_{j=l}^{n} \left\{\ddot{e}^{*}_{j-t}+ v\cdot e^*_{j-t}\right\} v^{j-t} & \text{für $t<l$ (Aufschubzeit)}\\ % \sum_{j=t}^{n} \left\{\ddot{e}^{*}_{j-t}+ v\cdot e^*_{j-t}\right\} v^{j-t} & \text{für $t\ge l$ (Liquiditätsphase)} % \end{cases}\\ -% &= \ddot{e}_t^{*} + \left\{E^{*}_{\act[x]{n}}(t+1) + e_t^*\right\}\cdot v \cdot \begin{cases} +% &= \ddot{e}_t^{*} + \left\{E^{*}_\xn(t+1) + e_t^*\right\}\cdot v \cdot \begin{cases} % 1 & \text{für $t<l$ (Aufschubzeit)}\\ % p_{x+t} & \text{für $t\ge l$ (Liquiditätsphase)} % \end{cases} @@ -334,7 +336,7 @@ E12_{\act[x,y]{n}}(t) &= \ddot{e}_{t} + v \cdot p_{x+t} \cdot p_{y+t} \cdot \lef Analog zu (bei konstanter Rente) \begin{align*} - \ddot{a}^{(m)}_{\act[x]{n}} &= \ddot{a}_x^{(m)} - {}_np_x \cdot v^n \cdot \ddot{a}_{x+n}^{(m)}\\ + \ddot{a}^{(m)}_\xn &= \ddot{a}_x^{(m)} - {}_np_x \cdot v^n \cdot \ddot{a}_{x+n}^{(m)}\\ \ddot{a}_x^{(m)} &= \alpha(m) \ddot{a}_x - \beta(m) \end{align*} mit @@ -356,25 +358,25 @@ ergibt sich auch für allgemeine unterjährige Erlebenszahlungen $\ddot{e}_t$ ei \begin{align*} -A^{(m)}_{\act[x]{n}}(t) &= \ddot{e}_t \cdot \ddot{a}_{\act[x+t]{1}}^{(m)} + v \cdot p_{x+1} \cdot A^{(m)}_{\act[x]{n}}(t+1)\\ - &= \ddot{e}_t \cdot \left\{\alpha(m) - \beta(m) \cdot \left(1-p_{x+t} \cdot v\right)\right\} + v\cdot p_{x+t} \cdot A^{(m)}_{\act[x]{n}}(t+1) +A^{(m)}_\xn(t) &= \ddot{e}_t \cdot \ddot{a}_{\act[x+t]{1}}^{(m)} + v \cdot p_{x+1} \cdot A^{(m)}_\xn(t+1)\\ + &= \ddot{e}_t \cdot \left\{\alpha(m) - \beta(m) \cdot \left(1-p_{x+t} \cdot v\right)\right\} + v\cdot p_{x+t} \cdot A^{(m)}_\xn(t+1) \end{align*} \subsubsection{Nachschüssige $m$-tel jährliche Auszahlung der Erlebensleistungen} \begin{align*} -A^{(m)}_{\act[x]{n}}(t) &= e_t \cdot a_{\act[x+t]{1}}^{(m)} + v \cdot p_{x+t} \cdot A^{(m)}_{\act[x]{n}}(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)}_{\act[x]{n}}(t+1) +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} \begin{align*} -A^{(m)}_{\act[x]{n}}(t) = &\ddot{e}_t \cdot \left\{\alpha(m) - \beta(m) \cdot \left(1-p_{x+t} \cdot v\right)\right\} + \\ +A^{(m)}_\xn(t) = &\ddot{e}_t \cdot \left\{\alpha(m) - \beta(m) \cdot \left(1-p_{x+t} \cdot v\right)\right\} + \\ &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)}_{\act[x]{n}}(t+1) + &v\cdot p_{x+t} \cdot A^{(m)}_\xn(t+1) \end{align*} @@ -383,21 +385,21 @@ A^{(m)}_{\act[x]{n}}(t) = &\ddot{e}_t \cdot \left\{\alpha(m) - \beta(m) \cdot \ \subsection{Ablebensbarwert} \begin{align*} -A_{\act[x]{n}}(t) &= \sum_{j=t}^n {}_{j-t}p_{x+t} \cdot q_{x+j} \cdot v^{j-t+1} \cdot a_{j} \\ - &= q_{x+t} \cdot v \cdot a_t + p_{x+t} \cdot v \cdot A_{\act[x]{n}}(t+1)\\ +A_\xn(t) &= \sum_{j=t}^n {}_{j-t}p_{x+t} \cdot q_{x+j} \cdot v^{j-t+1} \cdot a_{j} \\ + &= q_{x+t} \cdot v \cdot a_t + p_{x+t} \cdot v \cdot A_\xn(t+1)\\ % \intertext{prämienfreier Ablebensbarwert:} -A^{(prf)}_{\act[x]{n}}(t) &= q_{x+t} \cdot v \cdot a^{(prf.)}_t + p_{x+t} \cdot v \cdot A^{(prf.)}_{\act[x]{n}}(t+1)\\ +A^{(prf)}_\xn(t) &= q_{x+t} \cdot v \cdot a^{(prf.)}_t + p_{x+t} \cdot v \cdot A^{(prf.)}_\xn(t+1)\\ % \intertext{Prämienrückgewähr} -A^{(RG)}_{\act[x]{n}}(t) &= q_{x+t} \cdot v \cdot a^{(RG)}_t + p_{x+t} \cdot v \cdot A^{(RG)}_{\act[x]{n}}(t+1)\\ +A^{(RG)}_\xn(t) &= q_{x+t} \cdot v \cdot a^{(RG)}_t + p_{x+t} \cdot v \cdot A^{(RG)}_\xn(t+1)\\ % \end{align*} \subsection{Leistungsbarwert} \begin{align*} -BW^L_{\act[x]{n}}(t) &= E_{\act[x]{n}}(t) + A_{\act[x]{n}}(t) + (1+\rho^{RG}) \cdot A^{(RG)}_{\act[x]{n}}(t)\cdot BP_{\act[x]{n}} +BW^L_\xn(t) &= E_\xn(t) + A_\xn(t) + (1+\rho^{RG}) \cdot A^{(RG)}_\xn(t)\cdot BP_\xn \end{align*} @@ -406,25 +408,25 @@ BW^L_{\act[x]{n}}(t) &= E_{\act[x]{n}}(t) + A_{\act[x]{n}}(t) + (1+\rho^{RG}) \c \begin{align*} \intertext{Abschlusskostenbarwerte:} -AK^{(VS)}_{\act[x]{n}}(t) &= \alpha^{VS}_{t} + v \cdot p_{x+t} \cdot AK^{(VS)}_{\act[x]{n}}(t+1) \\ -AK^{(PS)}_{\act[x]{n}}(t) &= \alpha^{PS}_{t} + v \cdot p_{x+t} \cdot AK^{(PS)}_{\act[x]{n}}(t+1) \\ -AK^{(BP)}_{\act[x]{n}}(t) &= \alpha^{BP}_{t}+\alpha_{3a,t} + v \cdot p_{x+t} \cdot AK^{(BP)}_{\act[x]{n}}(t+1) \\ +AK^{(VS)}_\xn(t) &= \alpha^{VS}_{t} + v \cdot p_{x+t} \cdot AK^{(VS)}_\xn(t+1) \\ +AK^{(PS)}_\xn(t) &= \alpha^{PS}_{t} + v \cdot p_{x+t} \cdot AK^{(PS)}_\xn(t+1) \\ +AK^{(BP)}_\xn(t) &= \alpha^{BP}_{t}+\alpha_{3a,t} + v \cdot p_{x+t} \cdot AK^{(BP)}_\xn(t+1) \\ \intertext{Zillmerkostenbarwerte:} -ZK^{(VS)}_{\act[x]{n}}(t) &= z^{VS}_{t} + v \cdot p_{x+t} \cdot ZK^{(VS)}_{\act[x]{n}}(t+1) \\ -ZK^{(PS)}_{\act[x]{n}}(t) &= z^{PS}_{t} + v \cdot p_{x+t} \cdot ZK^{(PS)}_{\act[x]{n}}(t+1) \\ +ZK^{(VS)}_\xn(t) &= z^{VS}_{t} + v \cdot p_{x+t} \cdot ZK^{(VS)}_\xn(t+1) \\ +ZK^{(PS)}_\xn(t) &= z^{PS}_{t} + v \cdot p_{x+t} \cdot ZK^{(PS)}_\xn(t+1) \\ \intertext{Inkassokostenbarwerte:} -IK_{\act[x]{n}}(t) &= \beta^{BP}_{t} + v \cdot p_{x+t} \cdot IK_{\act[x]{n}}(t+1) \\ +IK_\xn(t) &= \beta^{BP}_{t} + v \cdot p_{x+t} \cdot IK_\xn(t+1) \\ \intertext{Verwaltungskostenbarwerte:} -VK^{(VS)}_{\act[x]{n}}(t) &= \gamma^{VS}_{t} + v \cdot p_{x+t} \cdot VK^{(VS)}_{\act[x]{n}}(t+1) \\ -VK^{(PS)}_{\act[x]{n}}(t) &= \gamma^{PS}_{t} + v \cdot p_{x+t} \cdot VK^{(PS)}_{\act[x]{n}}(t+1) \\ -VK^{frei}_{\act[x]{n}}(t) &= \gamma^{VS,frei}_{t} + v \cdot p_{x+t} \cdot VK^{frei}_{\act[x]{n}}(t+1) \\ +VK^{(VS)}_\xn(t) &= \gamma^{VS}_{t} + v \cdot p_{x+t} \cdot VK^{(VS)}_\xn(t+1) \\ +VK^{(PS)}_\xn(t) &= \gamma^{PS}_{t} + v \cdot p_{x+t} \cdot VK^{(PS)}_\xn(t+1) \\ +VK^{frei}_\xn(t) &= \gamma^{VS,frei}_{t} + v \cdot p_{x+t} \cdot VK^{frei}_\xn(t+1) \\ \end{align*} \subsection{Darstellung der Barwerte in Vektor-/Matrixform} Die Leistungs- und Kostenbarwerte können (wie auch die Cashflows zu einem Zeitpunkt) in Matrixform dargestellt werden (aus Gründen der Übersichtlichkeit wird hier bei -allen Termen der Subscript ${\act[x]{n}}$ unterlassen): +allen Termen der Subscript $\xn$ unterlassen): \begin{align*} \overrightarrow{BW}^L(t) &= \left( \begin{matrix} @@ -455,38 +457,70 @@ VK^{frei}(t) & - & -\\ \section{Prämien} -Nettoprämie: +\subsection{Nettoprämie:} \begin{align*} -NP_{\act[x]{n}} &= \frac{E_{\act[x]{n}}(0) + A_{\act[x]{n}}(0) + \left(1+\rho^{RG}\right) \cdot A^{(RG)}_{\act[x]{n}}(0) \cdot BP_{\act[x]{n}}}{P_{\act[x]{n}}(0)} \cdot \left(1+\rho\right)\\ -% -\intertext{Zillmerprämie (gezillmerte Nettoprämie):} +NP_\xn &= \frac{E_\xn(0) + A_\xn(0) + \left(1+\rho^{RG}\right) \cdot A^{(RG)}_\xn(0) \cdot BP_\xn}{P_\xn(0)} \cdot \left(1+\rho\right) +\end{align*} + + +\subsection{Zillmerprämie (gezillmerte Nettoprämie):} % TODO: Sind die beta-Kosten proportional zu BP oder zur ZP??? I.e. ist \beta im Nenner mit BP oder im Zähler? -ZP_{\act[x]{n}} &= +\begin{align*} +ZP_\xn &= +\frac{% + NP_\xn\cdot P_\xn(0) + + ZK^{(VS)}_\xn(0) + + ZK^{(PS)}_\xn(0) \cdot BP_\xn \cdot PS + + ZK^{(BP)}_\xn(0)\cdot BP_\xn +% +}{P_\xn(0)} +\end{align*} + +Varianten: +\begin{itemize} + \item % REFERENCE: GE Tarife! + $\beta$- und $\gamma$-Kosten auch in die Zillmerprämie eingerechnet. Einziger Unterschied zur Bruttoprämie ist dann, dass nur die Zillmerkosten statt der $\alpha$-Kosten aufgeteilt werden. +\begin{align*} +ZP_\xn &= \left[% - NP_{\act[x]{n}}\cdot P_{\act[x]{n}}(0) + - \left(ZK^{(VS)}_{\act[x]{n}}(0) + IK^{(VS)}_{\act[x]{n}}(0) + VK^{(VS)}_{\act[x]{n}}(0)\right) + \right.\\ - &\qquad\left.\left(ZK^{(PS)}_{\act[x]{n}}(0) + IK^{(PS)}_{\act[x]{n}}(0) + VK^{(PS)}_{\act[x]{n}}(0)\right) \cdot BP_{\act[x]{n}} \cdot PS + \right.\\ - &\qquad\left.\left(ZK^{(BP)}_{\act[x]{n}}(0) + IK^{(BP)}_{\act[x]{n}}(0) + VK^{(BP)}_{\act[x]{n}}(0)\right) \cdot BP_{\act[x]{n}} + NP_\xn\cdot P_\xn(0) + + \left(ZK^{(VS)}_\xn(0) + IK^{(VS)}_\xn(0) + VK^{(VS)}_\xn(0)\right) + \right.\\ + &\qquad\left.\left(ZK^{(PS)}_\xn(0) + IK^{(PS)}_\xn(0) + VK^{(PS)}_\xn(0)\right) \cdot BP_\xn \cdot PS + \right.\\ + &\qquad\left.\left(ZK^{(BP)}_\xn(0) + IK^{(BP)}_\xn(0) + VK^{(BP)}_\xn(0)\right) \cdot BP_\xn % \right] / % -\left(P_{\act[x]{n}}(0)\right) -% -\intertext{Bruttoprämie:} -BP_{\act[x]{n}} &= \frac% +\left(P_\xn(0)\right) +\end{align*} + + \item % REFERENCE: ME + Prämienrückgewähr proportional zu Zillmerprämie (für Berechnung der Zillmerprämie): +\begin{align*} +ZP_\xn &= \frac{E_\xn(0) + A_\xn(0) + \left(1+\rho^{RG}\right) \cdot A^{(RG)}_\xn(0) \cdot ZP_\xn}{P_\xn(0)} \cdot \left(1+\rho\right)\\ +ZP_\xn &= \frac{E_\xn(0) + A_\xn(0) + \cdot ZP_\xn}{P_\xn(0) - \left(1+\rho^{RG}\right)\cdot A^{(RG)}_\xn(0)\cdot \left(1+\rho\right)} \cdot \left(1+\rho\right) +\end{align*} + +\end{itemize} + + +\subsection{Bruttoprämie:} +\begin{align*} +BP_\xn &= \frac% { - \left(E_{\act[x]{n}}(0) + A_{\act[x]{n}}(0)\right)\cdot\left(1+\rho\right) + - \left( AK^{(VS)}_{\act[x]{n}}(0) + IK^{(VS)}_{\act[x]{n}}(0) + VK^{(VS)}_{\act[x]{n}}(0) \right) + \left(E_\xn(0) + A_\xn(0)\right)\cdot\left(1+\rho\right) + + \left( AK^{(VS)}_\xn(0) + IK^{(VS)}_\xn(0) + VK^{(VS)}_\xn(0) \right) }% { - P_{\act[x]{n}}(0) - - A^{(RG)}_{\act[x]{n}} \left(1+\rho^{RG}\right) \left(1+\rho\right) - - AK^{(BP)}_{\act[x]{n}} - IK^{(BP)}_{\act[x]{n}} - VK^{(BP)}_{\act[x]{n}} - - \left(AK^{(PS)}_{\act[x]{n}} + IK^{(PS)}_{\act[x]{n}} + VK^{(PS)}_{\act[x]{n}}\right) PS + P_\xn(0) - + A^{(RG)}_\xn \left(1+\rho^{RG}\right) \left(1+\rho\right) - + AK^{(BP)}_\xn - IK^{(BP)}_\xn - VK^{(BP)}_\xn - + \left(AK^{(PS)}_\xn + IK^{(PS)}_\xn + VK^{(PS)}_\xn\right) PS } -\intertext{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.} +\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. % -\intertext{Ablebensleistung im Jahr $t$:} -Abl(t) &= \left\{a_t + a^{(RG)}_t \cdot BP_{\act[x]{n}}\right\} \cdot VS +\subsection{Ablebensleistung im Jahr $t$:} +\begin{align*} +Abl(t) &= \left\{a_t + a^{(RG)}_t \cdot BP_\xn\right\} \cdot VS \end{align*} \subsection{Koeffizienten in Vektorschreibweise} @@ -500,20 +534,20 @@ Für die Berechnung der Prämien können die Koeffizienten der jeweiligen Barwer \begin{tabular}{||ll|c|c||}\hline\hline & & Leistungen & Kosten \\ \hline\hline Terme & & - $\begin{benarray}P_{\act[x]{n}}(t) & E^{Gar}_{\act[x]{n}}(t) & E_{\act[x]{n}}(t) & A_{\act[x]{n}}(t) & A_{\act[x]{n}}^{(RG)}(t)\end{benarray}$ + $\begin{benarray}P_\xn(t) & E^{Gar}_\xn(t) & E_\xn(t) & A_\xn(t) & A_\xn^{(RG)}(t)\end{benarray}$ & $\begin{costarray} -AK^{(VS)}_{\act[x]{n}}(t) & AK^{(PS)}_{\act[x]{n}}(t) & AK^{(BP)}_{\act[x]{n}}(t) \\ -ZK^{(VS)}_{\act[x]{n}}(t) & ZK^{(PS)}_{\act[x]{n}}(t) & -\\ -- & - & IK_{\act[x]{n}}(t) \\ -VK^{(VS)}_{\act[x]{n}}(t) & VK^{(PS)}_{\act[x]{n}}(t) & -\\ -VK^{frei}_{\act[x]{n}}(t) & - & -\\ +AK^{(VS)}_\xn(t) & AK^{(PS)}_\xn(t) & AK^{(BP)}_\xn(t) \\ +ZK^{(VS)}_\xn(t) & ZK^{(PS)}_\xn(t) & -\\ +- & - & IK_\xn(t) \\ +VK^{(VS)}_\xn(t) & VK^{(PS)}_\xn(t) & -\\ +VK^{frei}_\xn(t) & - & -\\ \end{costarray}$ \\\hline\hline Nettoprämie & Zähler & - $\begin{benarray}0 & 1+\rho & 1+\rho & 1+\rho & \left(1+\rho^{RG}\right)\cdot BP_{\act[x]{n}} \cdot \left(1+\rho\right)\end{benarray}$ + $\begin{benarray}0 & 1+\rho & 1+\rho & 1+\rho & \left(1+\rho^{RG}\right)\cdot BP_\xn \cdot \left(1+\rho\right)\end{benarray}$ & - \\ @@ -524,13 +558,13 @@ Nettoprämie & Zähler & Zillmerprämie & Zähler & - $\begin{benarray}0 & 1+\rho & 1+\rho & 1+\rho & \left(1+\rho^{RG}\right)\cdot BP_{\act[x]{n}} \cdot \left(1+\rho\right)\end{benarray}$ + $\begin{benarray}0 & 1+\rho & 1+\rho & 1+\rho & \left(1+\rho^{RG}\right)\cdot BP_\xn \cdot \left(1+\rho\right)\end{benarray}$ & $\begin{costarray} 0 & 0 & 0 \\ -1 & BP_{\act[x]{n}}\cdot PS & BP_{\act[x]{n}}\\ -1 & BP_{\act[x]{n}}\cdot PS & BP_{\act[x]{n}} \\ -1 & BP_{\act[x]{n}}\cdot PS & BP_{\act[x]{n}}\\ +1 & BP_\xn\cdot PS & BP_\xn\\ +\orZero{1} & \orZero{BP_\xn\cdot PS} & \orZero{BP_\xn} \\ +\orZero{1} & \orZero{BP_\xn\cdot PS} & \orZero{BP_\xn} \\ 0 & 0 & 0 \\ \end{costarray}$ \\ @@ -598,47 +632,77 @@ Bruttoprämie & Zähler & Vorgeschriebene Prämie: \begin{multline*} -PV_{\act[x]{n}} = \left\{ (BP_{\act[x]{n}} + oUZu - SuRa) \cdot VS \cdot (1-VwGew) + StkK\right\} \cdot \\ \left(1-PrRa-VwGew_{StkK}-PartnerRa\right)\cdot \frac{1+uz(k)}{k} \cdot (1+VSt) +PV_\xn = \left\{ (BP_\xn + oUZu - SuRa) \cdot VS \cdot (1-VwGew) + StkK\right\} \cdot \\ \left(1-PrRa-VwGew_{StkK}-PartnerRa\right)\cdot \frac{1+uz(k)}{k} \cdot (1+VSt) % \end{multline*} +\section{Absolute Cash-Flows und Barwerte} + +TODO + \pagebreak \section{Rückstellungen und Reserven} -Reserve prämienpflichtig: +\subsection{Deckungskapital / Reserve} + +\subsubsection{Nettodeckungskapital prämienpflichtig:} \begin{align*} -V_{\act[x]{n}}(t) &= \left\{BW^L_{\act[x]{n}}(t)\cdot(1+\rho) - ZP_{\act[x]{n}}\cdot P_{\act[x]{n}}(t)\right\} \cdot VS \\ -% -\intertext{Verwaltungskostenreserve:} -V^{VwK}_{\act[x]{n}}(t) &= \left\{ VK^{(VS)}_{\act[x]{n}}(t) - \left(\frac{VK^{(VS)}_{\act[x]{n}}(0)}{P_{\act[x]{n}}(0)}\right) \cdot P_{\act[x]{n}}(t)\right\} \cdot VS\\ -% -\intertext{Reserve prämienfrei:} -V^{frei}_{\act[x]{n}}(t) &= \left\{(E_{\act[x]{n}}(t) + A1_{\act[x]{n}}(t))\cdot\widetilde{VW} + TODO \cdot \min(f,m) \cdot BP_{\act[x]{n}}(x,n)\cdot VS\right\} \cdot (1+\rho) \\ -% -\intertext{Verwaltungskostenreserve prämienfrei:} -V^{WvK,frei}_{\act[x]{n}}(t) &= VK4_{\act[x]{n}}(t) \cdot \widetilde{VS} -% +V_\xn(t) &= \left\{BW^L_\xn(t)\cdot(1+\rho) - NP_\xn\cdot P_\xn(t)\right\} \cdot VS +\end{align*} + +\subsubsection{Zillmerreserve prämienpflichtig:} +TODO! +\begin{align*} +V_\xn(t) &= \left\{BW^L_\xn(t)\cdot(1+\rho) - ZP_\xn\cdot P_\xn(t)\right\} \cdot VS =\\ + &= \left\{BW^L_\xn(t)\cdot(1+\rho) - NP_\xn\cdot P_\xn(t) - ZK_\xn(0) \cdot BP_\xn(t) \cdot \frac{P_\xn(t)}{P_\xn(0)}\right\} \cdot VS \\ +\end{align*} + +\subsubsection{Reserve prämienpflichtig:} +Entspricht bei Zillmerung der Zillmerreserve +\begin{align*} +V_\xn(t) &= \left\{BW^L_\xn(t)\cdot(1+\rho) - ZP_\xn\cdot P_\xn(t)\right\} \cdot VS \\ +\end{align*} + +\subsubsection{Bruttoreserve prämienpflichtig:} + +\begin{align*} +V^{(b)}_\xn(t) &= \left\{BW^L_\xn(t)\cdot(1+\rho) + - ZP_\xn\cdot P_\xn(t)\right\} \cdot VS \\ +\end{align*} + +\subsection{Verwaltungskostenreserve:} +\begin{align*} +V^{VwK}_\xn(t) &= \left\{ VK^{(VS)}_\xn(t) - \left(\frac{VK^{(VS)}_\xn(0)}{P_\xn(0)}\right) \cdot P_\xn(t)\right\} \cdot VS\\ +\end{align*} + +\subsection{Reserve prämienfrei:} +\begin{align*} +V^{frei}_\xn(t) &= \left\{(E_\xn(t) + A1_\xn(t))\cdot\widetilde{VW} + TODO \cdot \min(f,m) \cdot BP_\xn(x,n)\cdot VS\right\} \cdot (1+\rho) \\ +\end{align*} + +\subsection{Verwaltungskostenreserve prämienfrei:} +\begin{align*} +V^{WvK,frei}_\xn(t) &= VK4_\xn(t) \cdot \widetilde{VS} \end{align*} \section{Spar- und Risikoprämie} \begin{equation*} - P_{\act[x]{n}}(t) = SP_{\act[x]{n}}(t) + RP_{\act[x]{n}}(t) + P_\xn(t) = SP_\xn(t) + RP_\xn(t) \end{equation*} \subsection{Sparprämie} \begin{align*} -SP_{\act[x]{n}}(t) &= V_{\act[x]{n}}(t+1) \cdot v - V_{\act[x]{n}}(t) + \left(\ddot{e}_t + v\cdot e_t\right)\cdot VS +SP_\xn(t) &= V_\xn(t+1) \cdot v - V_\xn(t) + \left(\ddot{e}_t + v\cdot e_t\right)\cdot VS \end{align*} \subsection{Risikoprämie} \begin{align*} -RP_{\act[x]{n}}(t) &= v\cdot q_{x+t} \cdot \left\{Abl(t) - V_{\act[x]{n}}(t+1)\right\} +RP_\xn(t) &= v\cdot q_{x+t} \cdot \left\{Abl(t) - V_\xn(t+1)\right\} \end{align*} @@ -661,56 +725,186 @@ $baf$ \dots & Bilanzabgrenzungsfaktor (Jahresanteil zwischen Abschlussdatum und % Bilanzabgrenzungsfaktor Bilanzreserve für Versicherungsleistungen: \begin{align*} -BilRes^{(L)}_{\act[x]{n}}(t) &= (1-baf)\cdot V_{\act[x]{n}}(t) + baf \cdot V_{\act[x]{n}}(t+1)\\ +BilRes^{(L)}_\xn(t) &= (1-baf)\cdot V_\xn(t) + baf \cdot V_\xn(t+1)\\ \intertext{Verwaltungskosten-Bilanzreserve:} -BilRes^{(VwK)}_{\act[x]{n}}(t) &= (1-baf)\cdot V^{(VwK)}_{\act[x]{n}}(t) + baf \cdot V^{(VwK)}_{\act[x]{n}}(t+1)\\ +BilRes^{(VwK)}_\xn(t) &= (1-baf)\cdot V^{(VwK)}_\xn(t) + baf \cdot V^{(VwK)}_\xn(t+1)\\ \intertext{Gesamte Bilanzreserve:} -BilRes_{\act[x]{n}}(t) &= BilRes^{(L)}_{\act[x]{n}}(t) + BilRes^{(VwK)}_{\act[x]{n}}(t)\\ +BilRes_\xn(t) &= BilRes^{(L)}_\xn(t) + BilRes^{(VwK)}_\xn(t)\\ \intertext{\subsection{prämienfrei} Bilanzreserve für Versicherungsleistungen, prämienfrei:} -BilRes^{(L),frei}_{\act[x]{n}}(t) &= (1-baf)\cdot V^{frei}_{\act[x]{n}}(t) + baf \cdot V^{frei}_{\act[x]{n}}(t+1)\\ +BilRes^{(L),frei}_\xn(t) &= (1-baf)\cdot V^{frei}_\xn(t) + baf \cdot V^{frei}_\xn(t+1)\\ \intertext{Verwaltungskosten-Bilanzreserve, prämienfrei:} -BilRes^{(VwK),frei}_{\act[x]{n}}(t) &= (1-baf)\cdot V^{VwK,frei}_{\act[x]{n}}(t) + baf \cdot V^{VwK,frei}_{\act[x]{n}}(t+1)\\ +BilRes^{(VwK),frei}_\xn(t) &= (1-baf)\cdot V^{VwK,frei}_\xn(t) + baf \cdot V^{VwK,frei}_\xn(t+1)\\ \intertext{Gesamte Bilanzreserve, prämienfrei:} -BilRes^{frei}_{\act[x]{n}}(t) &= BilRes^{(L),frei}_{\act[x]{n}}(t) + BilRes^{(VwK),frei}_{\act[x]{n}}(t)\\\ +BilRes^{frei}_\xn(t) &= BilRes^{(L),frei}_\xn(t) + BilRes^{(VwK),frei}_\xn(t)\\\ \end{align*} \pagebreak -\section{Rückkaufswerte} +\section{Prämienfreistellung und Rückkauf} -TODO +Verteilung der $\alpha$-Kosten auf $r$ Jahre für den Rückkauf bzw. die Vertragskonversion +ist nicht bei allen Tarifen oder in allen Jurisdiktionen vorgesehen. => FLAG +\subsection{Umrechnungsreserve} +Sowohl Prämienfreistellung als auch Rückkauf starten von der Umrechnungsreserve, die sich aus der +Zillmerreserve, den Kostenrückstellungen sowie der Verteilung der $\alpha$-Kosten auf 5 Jahre ergibt: +\begin{align*} + V_\xn^{Umr} &= \left(V_\xn(t) + V^{VwK}_\xn(t) + AbsKErh(t)\right)\cdot (1-VwGew(TODO)) +\end{align*} +wobei $AbsKErh(t)$ die anteilsmäßige Rückzahlung der Abschlusskosten bei Rückkauf innerhalb der ersten +$n(=5)$ Jahre gemäß \S 176 öVersVG bezeichnet: +\begin{align*} +AbskErh(t) &= \max\left(\sum_{j=0}^t Zillm(j) - \frac{t}{5} \sum_{j=0}^n Zillm(j), 0\right)&& \text{(Abschlusskostenerhöhungsbetrag)}\\ +Zillm(t) &= z^{(VS)}_t + z^{(BP)} \cdot BP_\xn + z^{(PS)}_t \cdot BP_\xn \cdot \sum_{j=0}^n pr_j &&\text{(Zillmerprämienanteil/-cashflow im Jahr $j$)} +\end{align*} -\pagebreak +Varianten: +\begin{itemize} + \item Verteilung auf 5 Jahre nicht linear ($t/5$), sondern als 5-jährige Leibrente bewertet, deren Rest noch ausständig ist. + \begin{align*} + AbskErh(t) &= \max\left(\sum_{j=0}^t Zillm(j) - \left(1-\frac{\ddot{a}_{\act[x+t]{r-t}}}{\ddot{a}_{\act[x]{r}}}\right) \frac{t}{5} \sum_{j=0}^n Zillm(j), 0\right) + \end{align*} + \item Bei zahlreichen Tarifen wird die Abschlusskostenerhöhung erst NACH dem Rückkaufsabschlag addidiert, + sodass diese Erhöhung nicht vom Abschlag betroffen ist => FLAG +\end{itemize} + +\subsection{Rückkaufswert (prämienpflichtig)} + +Zahlreiche Tarife sind NICHT rückkaufsfähig => FLAG + +\begin{align*} + Rkf(t) &= f(V_\xn^{Umr}, ...) +\end{align*} + +Die Abschläge von der Umrechnungsreserve auf den Rückkaufswert sind im Allgemeinen nicht +standardisiert, sondern variieren je nach Versicherungsunternehmen stark. Mögliche +Abschläge sind: + +\subsubsection*{Prozentualer Rückkaufsabschlag} +Prozentualer Abschlag auf die Umrechnungsreserve, z.B. $2\%$ oder $5\%$: +$RkfFakt=0.95$ +\begin{align*} +f(V_\xn^{Umr}, ...) = RkfFakt \cdot V_\xn^{Umr} \text{\quad mit $RkfFakt=0.98$ oder $0.95$} +\end{align*} + +\subsubsection*{Lineare Erhöhung des prozentualen Rückkaufsabschlags} +\begin{align*} +f(V_\xn^{Umr}, ...) &= RkfFakt(t) \cdot V_\xn^{Umr} \\ +RkfFakt(t) &= min(k_1 + t \cdot \delta k; k_2) \text{\quad mit z.B. $k_1=0.9$, $\delta k = 0.005$ und $k_2=0.98$} +\end{align*} +Alternativ: +\begin{equation*}% ME L12PLK3Z + RkfFakt(t) = \begin{cases} +0.95 & 1\leq t \leq 3\\ +0.95 + 0.003\cdot(t-3) & 3<t\leq 13\\ +0.98 & 13<t +\end{cases} +\end{equation*} + + +\subsubsection*{Prozentualer Abschlag mit Mindestabschlag} +% GW GDV1 +\begin{align*} +f(V_\xn^{Umr}, ...) &= min\left(0.95 \cdot V_\xn^{Umr}, Abl(t), V_\xn^{Umr}-0.15 \cdot BP_\xn \cdot VS\cdot (1-VwGew)\right) +\end{align*} + +% GW ER11 +\begin{align*} +f(V_\xn^{Umr}, ...) &= min\left(0.95 \cdot V_\xn^{Umr}, Abl(t)\right) +\end{align*} + + +\subsubsection*{Prozentualer Abschlag mit Mindestabschlag (Mindesttodesfallsumme als Grenze)} +% HP/GW: +\begin{align*} +f(V_\xn^{Umr}, ...) &= min(0.95 \cdot V_\xn^{Umr}, MTS(m,t)) \\ +MTS(m,t) &= ... +\end{align*} + + +\subsubsection*{Abschlag proportional zum Deckungskapital} +% GE +\begin{align*} +f(V_\xn^{Umr}, ...) &= V_\xn^{Umr}\cdot \left(s_f + \max(0.97-s_f, 0) \cdot \frac{V_\xn^{Umr}}{VS}\right) \\ +s_f &=\begin{cases}0.92 & \text{ für $t<\max(10, n-5)$}\\ 1 & \text{sonst}\end{cases} +\end{align*} + + +TODO: Weitere mögliche Rückkaufsabschläge rausfinden + +\subsection{Stornogebühr bei Rückkauf} + +Manche Tarife sehen eine fixe Stornogebühr bei Rückkauf (z.B. nur in den ersten 24 Monaten) vor: +% GW +\begin{equation*} + StoGeb = \min\left(\max\left(0.15 \cdot PV(x,n) \cdot\frac{pz}{1-uz(pz)} \cdot\frac1{1+VSt}, 30 \right), 300\right) +\end{equation*} +Ansonsten: $StoGeb=0$. -\section{Prämienfreistellung} -Der Vertrag wird zum Zeitpunkt $f$ prämienfrei gestellt, d.h. ab ZP $f$ wird keine Prämie mehr bezahlt, + +\subsection{Prämienfreistellung} + +Der Vertrag wird zum Zeitpunkt $f$ prämienfrei gestellt, d.h. ab $f$ wird keine Prämie mehr bezahlt, die Höhe des Versicherungsschutzes bestimmt sich aus dem zu $f$ vorhandenen Deckungskapital und den -Kostenreserven. Bei Prämienrückgewähr wird selbstverständlich nur die tatsächlich bezahlte Prämiensumme rückgewährt. +Kostenreserven (Umrechnungsreserve). Bei Prämienrückgewähr wird nur die tatsächlich bezahlte Prämiensumme rückgewährt. Aus -\begin{multline*} - V_{\act[x]{n}}(f) + V^{VwK}_{\act[x]{n}}(t) + AbskErh(f) \cdot VS \cdot \left(1-VwGew\right) = \\ - = BW^L_{\act[x]{n}}(f)\cdot\left(1+\rho\right)\cdot \widetilde{VS} + BW^{RG,frei}_{\act[x]{n}}(f)\cdot \left(1+\rho\right) \cdot BP_{\act[x]{n}} \cdot VS + VK^{frei}_{\act[x]{n}}(f) = - V^{frei}_{\act[x]{n}}(f) + V_{\act[x]{n}}^{VwK,frei}(f) -\end{multline*} +\begin{align*} + V_\xn^{Umr}(f) - StoGeb %= \\ + = \underbrace{BW^L_\xn(f)\cdot\left(1+\rho\right)\cdot \widetilde{VS} + BW^{RG,frei}_\xn(f)\cdot \left(1+\rho\right) \cdot BP_\xn \cdot VS}_{=V^{frei}_\xn(f)} + \underbrace{VK^{frei}_\xn(f)}_{=V_\xn^{VwK,frei}(f)} +% = V^{frei}_\xn(f) + V_\xn^{VwK,frei}(f) +\end{align*} mit \begin{align*} -% \intertext{Zillmerprämienanteil:} -Zillm(t) &= z^{(VS)}_t + z^{(BP)}_t \cdot BP_{\act[x]{n}} \cdot \sum_{j=0}^n pr_j &&\text{(Zillmerprämienanteil)}\\ -% \intertext{Abschlusskostenerhöhungsbetrag:} -AbskErh(t) &= \max\left(\sum_{j=0}^t Zillm(j) - \frac{t}{5} \sum_{j=0}^n Zillm(j), 0\right)&& \text{(Abschlusskostenerhöhungsbetrag)}\\ -% \intertext{Barwert zukünftiger Prämienrückgewähr:} -BW^{RG,frei}_{\act[x]{n}}(f) &= A^{(RG)}_{\act[x]{n}}(t) \cdot \min\left(f,m\right) && \text{(BW zukünftiger Prämienrückgewähr)} +BW^{RG,frei}_\xn(f) &= A^{(RG)}_\xn(t) \cdot \underbrace{\sum_{j=0}^{f-1} pr_j}_{\substack{=\min\left(f,m\right)\text{ bei }\\\text{lfd. konst. Prämie}}} && \text{(BW zukünftiger Prämienrückgewähr)} \end{align*} -ergibt sich die neue Versicherungssumme $\widetilde{VS}$ nach Prämienfreistellung: +ergibt sich die neue Versicherungssumme $\widetilde{VS}(f)$ nach Prämienfreistellung zum Zeitpunkt $f$: \begin{equation*} -\widetilde{VS} = \frac{V_{\act[x]{n}}(t) + V^{Vwk}_{\act[x]{n}}(t) + AbskErh(t)\cdot VS \cdot \left(1-VwGew\right) - BW^{RG,frei}_{\act[x]{n}}(f)\cdot (1+\rho)\cdot BP_{\act[x]{n}} \cdot VS}{E_{\act[x]{n}}(t)\cdot(1+\rho) + VK^{frei}_{\act[x]{n}}(t)} +\widetilde{VS}(f) = \frac + { V_\xn^{Umr}(f) - BW^{RG,frei}_\xn(f)\cdot (1+\rho)\cdot BP_\xn \cdot VS - StoGeb} + {BW^L_\xn(f)\cdot(1+\rho) + VK^{frei}_\xn(f)} \end{equation*} +\subsection{Reserven nach außerplanmäßiger Prämienfreistellung} + +\subsubsection*{Nettodeckungskapital außerplanmäßig Prämienfrei zu $f$} + +\begin{align*} +V^{(n),prf,f}_\xn(t) &= \left\{BW^{L,prf}_\xn(t)\cdot(1+\rho)\right\} \cdot \widetilde{VS(f)} +\end{align*} + +\subsubsection{Reserve außerplanmäßig prämienfrei:} + +\begin{align*} +V_\xn^{prf,f}(t) &= \left\{BW^{L,pr}_\xn(t)\cdot(1+\rho) + BW^{RG,frei,f}_{\act[x]{x}}(t)\right\} \cdot \widetilde{VS(f)} \\ +\end{align*} + +\subsection{Verwaltungskostenreserve außerplanmäßig prämienfrei:} +\begin{align*} +V^{VwK,prf,f}_\xn(t) &= \left\{ VK^{(VS), prf.}_\xn(t) + VK^{(PS),prf.}_\xn(t)\cdot PS(f)\right\} \cdot \widetilde{VS(f)}\\ +\end{align*} + + +TOCHECK: +\subsection{Reserve prämienfrei:} +\begin{align*} +V^{frei}_\xn(t) &= \left\{(E_\xn(t) + A1_\xn(t))\cdot\widetilde{VW} + TODO \cdot \min(f,m) \cdot BP_\xn(x,n)\cdot VS\right\} \cdot (1+\rho) \\ +\end{align*} + +\subsection{Verwaltungskostenreserve prämienfrei:} +\begin{align*} +V^{WvK,frei}_\xn(t) &= VK4_\xn(t) \cdot \widetilde{VS} +\end{align*} + +\subsection{Umrechnungsreserve außerplanmäßig prämienfrei} +\begin{align*} + V_\xn^{Umr, prf, f}(t) &= \left(V^{prf,f}_\xn(t) + V^{VwK,prf,f}_\xn(t)\right)\cdot (1-VwGew(TODO)) +\end{align*} + + -- GitLab