MATHHX B

MATHHX B

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{\morecmidrules }{}\) \(\newcommand {\specialrule }[3]{\hline }\) \(\newcommand {\addlinespace }[1][]{}\) \(\def \LWRsiunitxrangephrase { \protect \mbox {to (numerical range)} }\) \(\def \LWRsiunitxdecimal {.}\)

8.3 Annuitetslån

Et annuitetslån er et lån som tilbagebetales med faste ydelser med faste mellemrum. På den måde minder det om en annuitetsopsparing - det er bare et lån i stedet for en opsparing. Vi bruger betegnelserne:

.
\(y\)

Ydelsen. Det beløb vi betaler hver termin.

\(n\)

Antallet af ydelser.

\(r\)

Rentefoden dvs. renten pr. termin som decimaltal.

\(A_0\)

Det lånte beløb. Kaldes også hovedstolen eller nutidsværdien.

På en tidslinje:

(-tikz- diagram)

Øvelse 8.3.1

Ligesom ved annuitetsopsparing betegner \(n\) antallet af ydelser.

  • a) Hvor mange terminer er der i en annuitetslån med \(n\) ydelser?

Løsning 8.3.1

  • a) Der er også \(n\). I modsætning til annuitetsopsparing er der er nemlig ikke nogen ydelse i starten af tidslinjen, så der er lige så mange ydelser som der er mellemrum. Her er et eksempel hvor \(n=3\). Vi ser at der også er \(3\) rentetilskrivninger og dermed \(3\) terminer:

    (-tikz- diagram)

Der gælder følgende sætning:

  • Sætning 8.3.1
    For et annuitetslån kan nutidsværdien \(A_0\) bestemmes ved:

    \[A_0=y\cdot \frac {1-(1+r)^{-n}}{r}\]

Øvelse 8.3.2

Antag at vi har et annuitetslån med \(y=300\), \(n=5\) og \(r=0{,}012\).

  • a) Bestem nutidsværdien

Løsning 8.3.2

  • a) \(A_0=1447{,}48\)

Øvelse 8.3.3

En dame låner hele købesummen til et hus. Hun afbetaler med \(23.163{,}61\) kr., hver måned i \(30\) år (i alt \(360\) ydelser). Rentefoden er \(0{,}1\%\)

  • a) Hvad kostede huset?

Løsning 8.3.3

  • a) \(7\) millioner.

Ligesom ved annuitetsopsparing findes formlen for annuitetslån i flere versioner:

\(\begin {array}{|c|c|c|c|c|} \hline A_0 & y & r & n\\ \hline A_0=y\cdot \frac {1-(1+r)^{-n}}{r} & y=\frac {A_0\cdot r}{1-(1+r)^{-n}} & \text {findes ikke} & n=-\frac {\ln \big (1-\frac {A_0\cdot r}{y}\big )}{\ln (1+r)}\\ \hline \end {array}\)

Det ses at der ikke er nogen formel for \(r\). Her er du igen nødt til at opstille en ligning beregne løsningen i GeoGebra (tilsvarende til det vi gjorde for renten i en annuitetsopsparing i eksempel 8.2.2 ). Du kan også bruge Excel (se afsnit 8.5).

Øvelse 8.3.4

Brøndby IF låner 18 mio kr. af Jan Bech Andersen. Brøndby vælger at afdrage med et annuitetslån over 9 måneder med en månedlig ydelse på \(2.532.421{,}44\) kr.

  • a) Bestem den månedlige rente.

Løsning 8.3.4

  • a) \(5\%\)

Øvelse 8.3.5

En mand køber bil til \(450.000\) kr. Han skal betale bilen af med kvartårlige ydelser over \(10\) år (\(40\) ydelser) til en kvartårlig rente på \(1{,}5\%\)

  • a) Bestem hvor meget han skal betale tilbage hvert kvartal.

Løsning 8.3.5

  • a) \(y=15042{,}20\) kr.

Øvelse 8.3.6

En elev går amok og ødelægger inventaret i sit klasseværelse. Rektor beslutter at eleven skal erstatte det han har ødelagt. Eleven har ødelagt for \(35.000\) kr. Da eleven ikke har så mange penge må eleven låne pengene af en lånehaj. Eleven skal hver måned betale \(771{,}68\) kr. af på lånet. Renten er \(2\%\) pr. måned.

  • a) Bestem antallet af ydelser.

  • b) Bestem hvor mange år der går før eleven har fået tilbagebetalt lånet.

Løsning 8.3.6

  • a) \(120\)

  • b) \(10\) år