MATHHX B

MATHHX B

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8.2 Annuitetsopsparing

Forstil dig at du hver måned sætter \(1.000\) kr. ind på en konto, som bliver tilskrevet en fast månedlig rente. En sådan opsparing kaldes en annuitetsopsparing. Vi bruger følgende betegnelser:

.
\(y\)

Ydelsen. Det beløb vi indsætter hver termin.

\(n\)

Antallet af ydelser.

\(r\)

Rentefoden dvs. renten pr. termin som decimaltal

\(A_n\)

Slutbeløbet på kontoen efter \(n\) ydelser. Kaldes også fremtidsværdien.

På en tidslinje ser det således ud:

(-tikz- diagram)

Øvelse 8.2.1

I en annuitetsopsparing betegner \(n\) antallet af ydelser og altså ikke antallet af terminer som ved en kapitalfremskrivning.

  • a) Hvor mange terminer er der i en annuitetsopsparing?

Løsning 8.2.1

  • a) Der er \(n-1\). Hvis f.eks. \(n=2\) har vi to ydelser, men der er kun \(1\) termin fordi der kun er et mellemrum mellem ydelserne:

    (-tikz- diagram)

Det kunne godt se ud som om, det er lidt besværligt at beregne fremtidsværdien \(A_n\). Alle ydelserne ligger nemlig forskellige steder på tidslinjen, og derfor får de tilskrevet renter et forskelligt antal gange. Den første ydelse man indbetaler får tilskrevet en masse renter, mens det sidste slet ikke får tilskrevet nogle. Heldigvis har vi en formel til hjælp:

  • Sætning 8.2.1
    Fremtidsværdien \(A_n\) af en annuitet bestående af \(n\) ydelser \(y\) ved en rente på \(r\), er givet ved:

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

  • Eksempel 8.2.1
    Anais sætter \(1000\) kr. ind på en konto, hver termin i \(10\) terminer. Rentefoden er på \(1\%\). Vi vil bruge sætning 8.2.1 til at beregne, hvor mange penge Anais kan hæve efter \(10\) terminer. Vi skal altså finde \(A_n\) når \(y=1000\), \(n=10\) og \(r=0{,}01\). Vi bruger formlen for fremtidsværdien af en annuitet:

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

    og vi sætter vores tal ind

    \begin{align*} A_{10} & =1000\cdot \frac {(1+0{,}01)^{10}-1}{0{,}01}\\ & = 10462{,}21 \end{align*} Anais kan altså hæve \(10462{,}21\) kr. efter \(10\) terminer.

Øvelse 8.2.2

Lille Gysse opretter en konto i banken og sætter \(200\) kr. ind hver måned. Kontoen bliver tilskrevet en månedlig rente på \(0{,}5\%\).

  • a) Bestem rentefoden \(r\) i decimaltal.

  • b) Beregn hvor mange penge Lille Gysse har på sin konto efter \(27\) indbetalinger.

  • c) Hvor mange penge har Lille Gysse indbetalt i alt?

  • d) Hvor mange penge har Lille Gysse fået tilskrevet i renter?

Løsning 8.2.2

  • a) \(r=0{,}005\)

  • b) Lille Gysse har \(5766{,}07\) kr. på sin konto efter \(27\) indbetalinger.

  • c) Lille Gysse har indbetalt \(5400\) kr.

  • d) Lille Gysse har fået tilskrevet \(366{,}07\) kr. i rente

Ligesom ved kapitalfremskrivning findes formlen for annuitetsopsparing i flere versioner:

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

Det ses at der ikke er nogen formel for \(r\). Her er du nødt til at opstille en ligning og bruge GeoGebra til at løse den. Alternativt kan du bruge Excel (se afsnit 8.5).

  • Eksempel 8.2.2
    Vi vil bestemme renten for en annuitetsopsparing med \(A_n=5000\), \(n=24\) og \(y=201{,}24\). Vi tager udgangspunkt i formlen for \(A_n\) (vi kunne også have valgt en af de andre formler):

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

    Indsætter vi de kendte værdier får vi:

    \[5000=201{,}24\cdot \frac {(1+r)^{24}-1}{r}\]

    Vi åbner et CAS-vindue i GeoGebra og bruger kommandoen \(\verb |Beregn(ligning,Variabel)|\):

    (image)

    Som du kan se, har jeg indstillet mit GeoGebra til 10 decimaler (inde i indstillinger). Vi konkluderer at renten er \(0{,}3\%\).

Øvelse 8.2.3

En elev sparer op til gallafest. Eleven sparer \(3.000\) kr. op ved at betale \(249{,}86\) kr. ind på en konto hver måned i et år.

  • a) Hvad er den månedlige rente?

Løsning 8.2.3

  • a) \(0{,}01\%\)

Øvelse 8.2.4

Martins mor har lavet en børneopsparing til Martin. Hun har hver måned sat et hemmeligt beløb ind på kontoen som er blevet tilskrevet en månedlig rente på \(0{,}4\%\). Efter \(216\) indbetalinger på Martins 18-års fødselsdag hæver han hele sin børneopsparing. Han modtager \(34213{,}67\) kr.

  • a) Hvor mange penge satte Martins mor ind hver måned?

Løsning 8.2.4

  • a) Hun satte \(100\) kr. ind hver måned.

Øvelse 8.2.5

Michael indsætter hvert kvartal \(500\) kr. ind på en konto. Renten er \(3\%\) pr. kvartal

  • a) Efter nogle år har Michael \(46359{,}93\) kr. på kontoen. Hvor mange indbetalinger har Michael lavet? Hvor mange år er der gået?

Løsning 8.2.5

  • a) Michael har fortaget \(45\) indbetalinger dvs. der er gået \(11\) år.