Lösungsweg Übung 1: Gegeben sind: Höhe der Rate: 1.000 € Prozentsatz: 4% Zeit: 6 Jahre Gesucht ist: Der Endwert nach 6 Jahren 1. Zur Berechnung wird die Gleichung für das Guthaben nach 6 Jahren mit der vorschüssigen Rentenformel berechnet: ![](data:image/png;base64,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) 2. Einsetzen der Werte in die Formel ergibt: ![](data:image/png;base64,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) 3. Das Ergebnis lautet: ![](data:image/png;base64,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) Lösungsweg Übung 2: Gegeben sind: Höhe der Rate: 7.175 € Zeit: 15 Jahre Prozentsatz: 6,5% Gesucht ist: Der Barwert am Anfang der 15 Jahre 1. Zur Berechnung wird die Gleichung für das Guthaben am Anfang der 15 Jahre mit der vorschüssigen Barwertformel verwendet: ![](data:image/png;base64,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) 2. Einsetzen der Werte in die Formel ergibt: ![](data:image/png;base64,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) 3. Das Ergebnis lautet: ![](data:image/png;base64,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)
von Sandra, Steffi Quelle: https://www.tfh-wildau.de/baetjer/oldpage/Auf_B1/Prakt/Prakt_B2/Pipiale/Pipiale.htm |