MATHHX B

MATHHX B

\(\newcommand{\footnotename}{footnote}\) \(\def \LWRfootnote {1}\) \(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\) \(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\) \(\let \LWRorighspace \hspace \) \(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\) \(\newcommand {\mathnormal }[1]{{#1}}\) \(\newcommand \ensuremath [1]{#1}\) \(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \) \(\newcommand {\setlength }[2]{}\) \(\newcommand {\addtolength }[2]{}\) \(\newcommand {\setcounter }[2]{}\) \(\newcommand {\addtocounter }[2]{}\) \(\newcommand {\arabic }[1]{}\) \(\newcommand {\number }[1]{}\) \(\newcommand {\noalign }[1]{\text {#1}\notag \\}\) \(\newcommand {\cline }[1]{}\) \(\newcommand {\directlua }[1]{\text {(directlua)}}\) \(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\) \(\newcommand {\protect }{}\) \(\def \LWRabsorbnumber #1 {}\) \(\def \LWRabsorbquotenumber "#1 {}\) \(\newcommand 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#1.#2.#3\LWRsiunitxEND {\LWRsiunitxprintdecimalsubtwo #1,,\LWRsiunitxENDTWO \ifblank {#2}{}{{\LWRsiunitxdecimal }\LWRsiunitxprintdecimalsubtwo #2,,\LWRsiunitxENDTWO }}\) \(\newcommand {\LWRsiunitxprintdecimal }[1]{\LWRsiunitxprintdecimalsub #1...\LWRsiunitxEND }\) \(\def \LWRsiunitxnumplus #1+#2+#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxprintdecimal {#1}}{\ifblank {#1}{\LWRsiunitxprintdecimal {#2}}{\LWRsiunitxprintdecimal {#1}\unicode {x02B}\LWRsiunitxprintdecimal {#2}}}\LWRsiunitxdistribunit }\) \(\def \LWRsiunitxnumminus #1-#2-#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumplus #1+++\LWRsiunitxEND }{\ifblank {#1}{}{\LWRsiunitxprintdecimal {#1}}\unicode {x02212}\LWRsiunitxprintdecimal {#2}\LWRsiunitxdistribunit }}\) \(\def \LWRsiunitxnumpmmacro #1\pm #2\pm #3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumminus #1---\LWRsiunitxEND }{\LWRsiunitxprintdecimal {#1}\unicode {x0B1}\LWRsiunitxprintdecimal {#2}\LWRsiunitxdistribunit }}\) \(\def \LWRsiunitxnumpm #1+-#2+-#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumpmmacro #1\pm \pm \pm \LWRsiunitxEND }{\LWRsiunitxprintdecimal {#1}\unicode {x0B1}\LWRsiunitxprintdecimal {#2}\LWRsiunitxdistribunit }}\) \(\newcommand {\LWRsiunitxnumscientific }[2]{\ifblank {#1}{}{\ifstrequal {#1}{-}{-}{\LWRsiunitxprintdecimal {#1}\times }}10^{\LWRsiunitxprintdecimal {#2}}\LWRsiunitxdistribunit }\) \(\def \LWRsiunitxnumD #1D#2D#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumpm #1+-+-\LWRsiunitxEND }{\mathrm {\LWRsiunitxnumscientific {#1}{#2}}}}\) \(\def \LWRsiunitxnumd #1d#2d#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumD #1DDD\LWRsiunitxEND }{\mathrm {\LWRsiunitxnumscientific {#1}{#2}}}}\) \(\def \LWRsiunitxnumE #1E#2E#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumd #1ddd\LWRsiunitxEND }{\mathrm {\LWRsiunitxnumscientific {#1}{#2}}}}\) \(\def \LWRsiunitxnume #1e#2e#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumE #1EEE\LWRsiunitxEND }{\mathrm {\LWRsiunitxnumscientific {#1}{#2}}}}\) \(\def \LWRsiunitxnumx #1x#2x#3x#4\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnume #1eee\LWRsiunitxEND }{\ifblank {#3}{\LWRsiunitxnume #1eee\LWRsiunitxEND \times \LWRsiunitxnume #2eee\LWRsiunitxEND }{\LWRsiunitxnume #1eee\LWRsiunitxEND \times \LWRsiunitxnume #2eee\LWRsiunitxEND \times \LWRsiunitxnume #3eee\LWRsiunitxEND }}}\) \(\newcommand {\num }[2][]{\LWRsiunitxnumx #2xxxxx\LWRsiunitxEND }\) \(\newcommand {\si }[2][]{\mathrm {\gsubstitute {#2}{~}{\,}}}\) \(\def \LWRsiunitxSIopt #1[#2]#3{\def \LWRsiunitxdistribunit {\,\si {#3}}{#2}\num {#1}\def \LWRsiunitxdistribunit {}}\) \(\newcommand {\LWRsiunitxSI }[2]{\def \LWRsiunitxdistribunit {\,\si {#2}}\num {#1}\def \LWRsiunitxdistribunit {}}\) \(\newcommand {\SI }[2][]{\ifnextchar [{\LWRsiunitxSIopt {#2}}{\LWRsiunitxSI {#2}}}\) \(\newcommand {\numlist }[2][]{\text {#2}}\) \(\newcommand {\numrange }[3][]{\num {#2}\ \LWRsiunitxrangephrase \ \num {#3}}\) \(\newcommand {\SIlist }[3][]{\text {#2}\,\si {#3}}\) \(\newcommand {\SIrange }[4][]{\num {#2}\,#4\ \LWRsiunitxrangephrase \ \num {#3}\,#4}\) \(\newcommand {\tablenum }[2][]{\mathrm {#2}}\) \(\newcommand {\ampere }{\mathrm {A}}\) \(\newcommand {\candela }{\mathrm {cd}}\) \(\newcommand {\kelvin }{\mathrm {K}}\) \(\newcommand {\kilogram }{\mathrm {kg}}\) \(\newcommand {\metre }{\mathrm {m}}\) \(\newcommand {\mole }{\mathrm {mol}}\) \(\newcommand {\second }{\mathrm {s}}\) \(\newcommand {\becquerel }{\mathrm {Bq}}\) \(\newcommand {\degreeCelsius }{\unicode {x2103}}\) \(\newcommand {\coulomb }{\mathrm {C}}\) \(\newcommand {\farad }{\mathrm {F}}\) \(\newcommand {\gray }{\mathrm {Gy}}\) \(\newcommand {\hertz }{\mathrm {Hz}}\) \(\newcommand {\henry }{\mathrm {H}}\) \(\newcommand {\joule }{\mathrm {J}}\) \(\newcommand {\katal }{\mathrm {kat}}\) \(\newcommand {\lumen }{\mathrm {lm}}\) \(\newcommand {\lux }{\mathrm {lx}}\) \(\newcommand {\newton }{\mathrm {N}}\) \(\newcommand {\ohm }{\mathrm {\Omega }}\) \(\newcommand {\pascal }{\mathrm {Pa}}\) \(\newcommand {\radian }{\mathrm {rad}}\) \(\newcommand {\siemens }{\mathrm {S}}\) \(\newcommand {\sievert }{\mathrm {Sv}}\) \(\newcommand {\steradian }{\mathrm {sr}}\) \(\newcommand {\tesla }{\mathrm {T}}\) \(\newcommand {\volt }{\mathrm {V}}\) \(\newcommand {\watt }{\mathrm {W}}\) \(\newcommand {\weber }{\mathrm {Wb}}\) \(\newcommand {\day }{\mathrm {d}}\) \(\newcommand {\degree }{\mathrm {^\circ }}\) \(\newcommand {\hectare }{\mathrm {ha}}\) \(\newcommand {\hour }{\mathrm {h}}\) \(\newcommand {\litre }{\mathrm {l}}\) \(\newcommand {\liter }{\mathrm {L}}\) \(\newcommand {\arcminute }{^\prime }\) \(\newcommand {\minute }{\mathrm {min}}\) \(\newcommand {\arcsecond }{^{\prime \prime }}\) \(\newcommand {\tonne }{\mathrm {t}}\) \(\newcommand {\astronomicalunit }{au}\) \(\newcommand {\atomicmassunit }{u}\) \(\newcommand {\bohr }{\mathit {a}_0}\) \(\newcommand {\clight }{\mathit {c}_0}\) \(\newcommand {\dalton }{\mathrm {D}_\mathrm {a}}\) \(\newcommand {\electronmass }{\mathit {m}_{\mathrm {e}}}\) \(\newcommand {\electronvolt }{\mathrm {eV}}\) \(\newcommand 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}\) \(\newcommand {\pF }{\pico \farad }\) \(\newcommand {\K }{\mathrm {K}}\) \(\newcommand {\dB }{\mathrm {dB}}\) \(\newcommand {\kibi }{\mathrm {Ki}}\) \(\newcommand {\mebi }{\mathrm {Mi}}\) \(\newcommand {\gibi }{\mathrm {Gi}}\) \(\newcommand {\tebi }{\mathrm {Ti}}\) \(\newcommand {\pebi }{\mathrm {Pi}}\) \(\newcommand {\exbi }{\mathrm {Ei}}\) \(\newcommand {\zebi }{\mathrm {Zi}}\) \(\newcommand {\yobi }{\mathrm {Yi}}\) \(\let \unit \si \) \(\let \qty \SI \) \(\let \qtylist \SIlist \) \(\let \qtyrange \SIrange \) \(\let \numproduct \num \) \(\let \qtyproduct \SI \) \(\let \complexnum \num \) \(\newcommand {\complexqty }[3][]{(\complexnum {#2})\si {#3}}\) \(\newcommand {\mleft }{\left }\) \(\newcommand {\mright }{\right }\) \(\newcommand {\mleftright }{}\) \(\newcommand {\mleftrightrestore }{}\) \(\require {gensymb}\) \(\newcommand {\intertext }[1]{\text {#1}\notag \\}\) \(\let \Hat \hat \) \(\let \Check \check \) \(\let \Tilde \tilde \) \(\let \Acute \acute \) \(\let \Grave \grave \) 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{\morecmidrules }{}\) \(\newcommand {\specialrule }[3]{\hline }\) \(\newcommand {\addlinespace }[1][]{}\) \(\def \LWRsiunitxrangephrase { \protect \mbox {to (numerical range)} }\) \(\def \LWRsiunitxdecimal {.}\)

8.5 Finansiel regning i Excel

De finansielle størrelser, vi har lært om indtil videre, kan hurtigt regnes i Excel. Dog er der noget fortegnshalløjsa, man skal være opmærksom på. I Excel regnes alle udgående beløb (penge man betaler) negative, mens indgående beløb (penge man modtager) regnes positive. Hvis Excel ikke lige er din kop te kan du vel godt springe dette afsnit over. Bemærk dog, at hvis du skal finde renten i en annuitet, så kræver det GeoGebra eller Excel.

Her er en oversigt over, hvordan man bestemmer de forskellige finansielle størrelser i Excel. Jeg har været flink og skrevet minus på de størrelser som skal indtastes som negative. Nogle af formlerne giver et negativt resultat. Hvis du f.eks. bestemmer \(K_0\) ud fra \(K_n\) vil den give dig et negativt tal, fordi du skal indsætte \(K_0\) for at kunne hæve \(K_n\). Det betyder dog ikke at du skal angive facit med et minus. Det er bare Excel, der prøver at fortælle dig, at det penge du skal af med.

Kapitalfremskrivning
.
Slutkapital (\(K_n\)) =FV(\(r\);\(n\);\(0\);-\(K_0\))
Startkapital (\(K_0\)) =NV(\(r\);\(n\);\(0\);\(K_n\))
Rente (\(r\)) =RENTE(\(n\);\(0\);-\(K_0\);\(K_n\))
Antal terminer (\(n\)) =NPER(\(r\);\(0\);-\(K_0\);\(K_n\))
Annuitetsopsparing
.
Fremtidsværdi (\(A_n\)) =FV(\(r\);\(n\);-\(y\))
Ydelse (\(y\)) =YDELSE(\(r\);\(n\);\(0\);\(A_n\))
Rente (\(r\)) =RENTE(\(n\);-\(y\);\(0\);\(A_n\))
Antal ydelser (\(n\)) =NPER(\(r\);-\(y\);\(0\);\(A_n\))
Annuitetslån
.
Nutidsværdi (hovedstol) (\(A_0\)) =NV(\(r\);\(n\);-\(y\))
Ydelse (\(y\)) =YDELSE(\(r\);\(n\);\(A_0\))
Rente (\(r\)) =RENTE(\(n\);-\(y\);\(A_0\))
Antal ydelser (\(n\)) =NPER(\(r\);-\(y\);\(A_0\))
Restgæld
.
Restgæld efter t terminer (\(R_t\)) =FV(\(r\);\(t\);-\(y\);\(A_0\))
  • Eksempel 8.5.1
    Vi vil bestemme renten for en annuitetsopsparing med \(A_n=5000\), \(n=24\) og \(y=201{,}24\).

    Vi bruger Excel. Vi skriver i en celle:

    \[\verb |=RENTE(24;-201,24;0;5000)|\]

    Læg mærke til minusset foran fremtidsværdien.

    Mit Excel viser svaret som 0%:

    Jeg trykker på knappen for flere decimaler:

    (image)

    og får svaret 0,30%

Øvelse 8.5.1

Tulle låner \(3.000\) kr. i banken, som hun betaler tilbage med en månedlig ydelse på \(166{,}25\) kr. Renten er \(1\%\) pr. måned.

  • a) Bestem antallet af ydelser. Brug Excel.

  • b) Bestem restgælden efter \(10\) måneder. Brug Excel.

Løsning 8.5.1

  • a) \(20\)

  • b) \(1.574{,}52\) kr.

Øvelse 8.5.2

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? Brug Excel.

Løsning 8.5.2

  • a) \(0{,}01\%\)

Øvelse 8.5.3

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 vha. Excel den månedlige rente.

Løsning 8.5.3

  • a) \(5\%\)

Ekstra

Vi ser at der i virkeligheden kun er 4 forskellige funktioner, man skal kende i Excel. Når man ved hvad forkortelserne står for, kan man måske regne ud, hvordan man bruger de forskellige funktioner. Det var i hvert fald det jeg selv gjorde, da jeg skulle skrive dette afsnit.

.
FV Står for FremtidsVærdi
NV Står for NutidsVærdi
RENTE Hvad tror du?
NPER Number of PERiods, dvs. antal ydelser

Jeg anbefaler at du kigger på Excel-funktionerne igen efter du har læst afsnit 8.7. Så vil du nemlig have et bedre grundlag til at forstå dem. Formlen for restgæld er nok lidt svær at gennemskue, da den kræver en lidt dybere forståelse af restgæld.

Jeg har sat minus på funktionerne i tabellerne så de virker i praksis, men skal man bruge funktionerne formelt korrekt, afhænger placeringen af evt. minus-tegn af konteksten. Sætter man \(100\) kr. ind på en konto, til en rente på \(5\%\) om året, skal man regne beløbet, man kan hæve efter \(10\) år ved:

\[\verb |=FV(0,05;10;0;-100)|=162{,}89\]

Låner man \(100\) kr. til en årlig rente på \(5\%\), skal man regne gælden efter 10 år ved:

\[\verb |=FV(0,05;10;0;100)|= \textcolor {red}{-162{,}89}\]

Læg mærke til forskellen med minus-tegnet på de \(100\) kr. Ved opsparing skal man af med \(100\) kr., og derfor skal det være minus. Ved lån modtager man \(100\) kr. og derfor er det et positivt beløb. Vi ser også at facit giver noget positivt ved opsparing og negativt ved gæld. Det er igen fordi at vi kan modtage \(162{,}89\) kr. ved opsparing, mens at vi skal betale \(162{,}89\) kr. ved gæld.