Empruntsindivis - public.iutenligne.net

Définitions. Un exemple d’emprunt. Emprunt à annuités constantes. Emprunt à amortissements constants.

Emprunts indivisannuités constantes - amortissements constants - TEG et TAEG

A. Claeys

GEA - IUT A - Lille 1

Février 2013


Plan

1 Définitions.Emprunt indivis - annuité - amortissement.Annuité de fin de période.

2 Un exemple d’emprunt.Calcul de l’annuité.Tableau d’amortissement.Taux unique équivalent.Taux actuariel effectif global (TAEG).

3 Emprunt à annuités constantes.Formule des annuités fixes.Tableau d’amortissement.Progression des amortissements.Taux effectif global, taux actuariel effectif global (TEG - TAEG).

4 Emprunt à amortissements constants.Formule des amortissments.Tableau d’amortissement.


Plan






Définitions.

DéfinitionEmprunt indivis : emprunt contracté auprès d’un seul prêteur.

DéfinitionAnnuités : montants versés périodiquement pour le remboursement.

DéfinitionAmortissement : part de l’annuité qui rembourse le capital.

intérêt

amortissement

annuité de fin de période k


Définitions.




intérêt

amortissement



Définitions.




intérêt

amortissement



Définitions.




intérêt

amortissement



Plan






Annuité de fin de période.

Dans ce cours, on considère le cas des annuités de fin de période :

Le débiteur paie la première annuité une période après le versement ducapital.

L’intérêt est calculé sur le capital restant dû en début de période.












Plan






Calcul de l’annuité.

ExempleAndre emprunte 1000=C remboursable en 2 annuités égales à terme échu.Année 1 : taux effectif annuel 8%. Année 2 : taux effectif annuel 5%.

axe du temps :années écoulées : 0 1 2

entrées :

1000

X X

sorties : A une date donnée, les capitaux entrés doivent être équivalents aux capitaux sortis.

X = 553,17=C.





entrées :

1000

X X


X = 553,17=C.





entrées :

1000

X X


X = 553,17=C.





entrées :

1000

X X


X = 553,17=C.





entrées :

1000

X X


X = 553,17=C.




taux annuel : 8% taux annuel : 5%axe du temps :

années écoulées : 0 1 2

entrées :

1000

X X


X = 553,17=C.






entrées :

1000

X X


X = 553,17=C.






entrées :

1000

X X


Equation de valeur à t = 2 :

X = 553,17=C.






entrées :

1000

X X



X +X (1+0,05) = 1000(1+0,08)(1+0,05) .

X = 553,17=C.






entrées :

1000

X X



X +X (1+0,05) = 1000(1+0,08)(1+0,05) .

X = 553,17=C.






entrées :

1000

X X



X +X (1+0,05) = 1000(1+0,08)(1+0,05) .

X = 553,17=C.






entrées :

1000

X X



X +X (1+0,05) = 1000(1+0,08)(1+0,05) .

(1+1,05)X = 1000×1,08×1,05

X = 553,17=C.






entrées :

1000

X X



X +X (1+0,05) = 1000(1+0,08)(1+0,05) .

(1+1,05)X = 1000×1,08×1,05

X = 1000×1,08×1,052,05

X = 553,17=C.






entrées :

1000

X X



X +X (1+0,05) = 1000(1+0,08)(1+0,05) .

(1+1,05)X = 1000×1,08×1,05

X = 1000×1,08×1,052,05

X = 553,17=C.


Plan






Tableau d’amortissement.

Remboursements du capital et des intérêts à chaque période.

DateCapital dû en

début de périodeAnnuité Amortissement Intérêt

Capital dû enfin de période

1 D0 = 1000,00=C a1 = 553,17=C A1 = 473,17=C I1 = 80=C D1 = 526,83=C

2 D1 = 526,83=C a2 = 553,17=C A2 = 526,83=C I2 = 26,34=C D2 = 0,00=C

Dette : D1 = 1000−473,17 = 526,83=C. Reste dû : D2 = 526,83−526,83 = 0,00=C.




DateCapital dû en



1 D0 = 1000,00=C a1 = 553,17=C A1 = 473,17=C I1 = 80=C D1 = 526,83=C

2 D1 = 526,83=C a2 = 553,17=C A2 = 526,83=C I2 = 26,34=C D2 = 0,00=C

A la date 1 :





DateCapital dû en



1 D0 = 1000,00=C a1 = 553,17=C A1 = 473,17=C I1 = 80=C D1 = 526,83=C

2 D1 = 526,83=C a2 = 553,17=C A2 = 526,83=C I2 = 26,34=C D2 = 0,00=C

A la date 1 :

Dette : D0 = 1000=C.





DateCapital dû en



1 D0 = 1000,00=C a1 = 553,17=C A1 = 473,17=C I1 = 80=C

×0,08

D1 = 526,83=C

2 D1 = 526,83=C a2 = 553,17=C A2 = 526,83=C I2 = 26,34=C D2 = 0,00=C

A la date 1 :

Dette : D0 = 1000=C.

Intérêt : I1 = 1000×0,08 = 80=C.





DateCapital dû en



1 D0 = 1000,00=C a1 = 553,17=C A1 = 473,17=C I1 = 80=C D1 = 526,83=C

2 D1 = 526,83=C a2 = 553,17=C A2 = 526,83=C I2 = 26,34=C D2 = 0,00=C

A la date 1 :

Dette : D0 = 1000=C.

Intérêt : I1 = 1000×0,08 = 80=C.

Annuité : a1 = 553,17=C.





DateCapital dû en



1 D0 = 1000,00=C a1 = 553,17=C A1 = 473,17=C

−

I1 = 80=C D1 = 526,83=C

2 D1 = 526,83=C a2 = 553,17=C A2 = 526,83=C I2 = 26,34=C D2 = 0,00=C

A la date 1 :

Dette : D0 = 1000=C.

Intérêt : I1 = 1000×0,08 = 80=C.

Annuité : a1 = 553,17=C.

Amortissement : A1 = 553,17−80 = 473,17=C.





DateCapital dû en



1 D0 = 1000,00=C a1 = 553,17=C A1 = 473,17=C I1 = 80=C D1 = 526,83=C

−2 D1 = 526,83=C a2 = 553,17=C A2 = 526,83=C I2 = 26,34=C D2 = 0,00=C

A la date 1 :

Dette : D0 = 1000=C.

Intérêt : I1 = 1000×0,08 = 80=C.

Annuité : a1 = 553,17=C.


Reste dû : D1 = 1000−473,17 = 526,83=C. Reste dû : D2 = 526,83−526,83 = 0,00=C.




DateCapital dû en



1 D0 = 1000,00=C a1 = 553,17=C A1 = 473,17=C I1 = 80=C D1 = 526,83=C

2 D1 = 526,83=C a2 = 553,17=C A2 = 526,83=C I2 = 26,34=C D2 = 0,00=C

A la date 1 :

Dette : D0 = 1000=C.

Intérêt : I1 = 1000×0,08 = 80=C.

Annuité : a1 = 553,17=C.


Reste dû : D1 = 1000−473,17 = 526,83=C.

A la date 2 :

Reste dû : D2 = 526,83−526,83 = 0,00=C.




DateCapital dû en



1 D0 = 1000,00=C a1 = 553,17=C A1 = 473,17=C I1 = 80=C D1 = 526,83=C

2 D1 = 526,83=C a2 = 553,17=C A2 = 526,83=C I2 = 26,34=C D2 = 0,00=C

A la date 1 :

Dette : D0 = 1000=C.

Intérêt : I1 = 1000×0,08 = 80=C.

Annuité : a1 = 553,17=C.


Reste dû : D1 = 1000−473,17 = 526,83=C.

A la date 2 :

Dette : D1 = 526,83=C.

Reste dû : D2 = 526,83−526,83 = 0,00=C.




DateCapital dû en



1 D0 = 1000,00=C a1 = 553,17=C A1 = 473,17=C I1 = 80=C D1 = 526,83=C

2 D1 = 526,83=C a2 = 553,17=C A2 = 526,83=C I2 = 26,34=C

×0,05

D2 = 0,00=C

A la date 1 :

Dette : D0 = 1000=C.

Intérêt : I1 = 1000×0,08 = 80=C.

Annuité : a1 = 553,17=C.


Reste dû : D1 = 1000−473,17 = 526,83=C.

A la date 2 :

Dette : D1 = 526,83=C.

Intérêt : I2 = 526,83×0,05 = 26,34=C.

Reste dû : D2 = 526,83−526,83 = 0,00=C.




DateCapital dû en



1 D0 = 1000,00=C a1 = 553,17=C A1 = 473,17=C I1 = 80=C D1 = 526,83=C

2 D1 = 526,83=C a2 = 553,17=C A2 = 526,83=C I2 = 26,34=C D2 = 0,00=C

A la date 1 :

Dette : D0 = 1000=C.

Intérêt : I1 = 1000×0,08 = 80=C.

Annuité : a1 = 553,17=C.


Reste dû : D1 = 1000−473,17 = 526,83=C.

A la date 2 :

Dette : D1 = 526,83=C.

Intérêt : I2 = 526,83×0,05 = 26,34=C.

Annuité : a2 = 553,17=C.

Reste dû : D2 = 526,83−526,83 = 0,00=C.




DateCapital dû en



1 D0 = 1000,00=C a1 = 553,17=C A1 = 473,17=C I1 = 80=C D1 = 526,83=C

2 D1 = 526,83=C a2 = 553,17=C A2 = 526,83=C

−

I2 = 26,34=C D2 = 0,00=C

A la date 1 :

Dette : D0 = 1000=C.

Intérêt : I1 = 1000×0,08 = 80=C.

Annuité : a1 = 553,17=C.


Reste dû : D1 = 1000−473,17 = 526,83=C.

A la date 2 :

Dette : D1 = 526,83=C.

Intérêt : I2 = 526,83×0,05 = 26,34=C.

Annuité : a2 = 553,17=C.

Amortissement : A2 = 553,17−26,34 = 526,83=C.

Reste dû : D2 = 526,83−526,83 = 0,00=C.




DateCapital dû en



1 D0 = 1000,00=C a1 = 553,17=C A1 = 473,17=C I1 = 80=C D1 = 526,83=C

2 D1 = 526,83=C a2 = 553,17=C A2 = 526,83=C I2 = 26,34=C D2 = 0,00=C

−

A la date 1 :

Dette : D0 = 1000=C.

Intérêt : I1 = 1000×0,08 = 80=C.

Annuité : a1 = 553,17=C.


Reste dû : D1 = 1000−473,17 = 526,83=C.

A la date 2 :

Dette : D1 = 526,83=C.

Intérêt : I2 = 526,83×0,05 = 26,34=C.

Annuité : a2 = 553,17=C.


Reste dû : D2 = 526,83−526,83 = 0,00=C.




DateCapital dû en



1 D0 = 1000,00=C a1 = 553,17=C A1 = 473,17=C I1 = 80=C D1 = 526,83=C

2 D1 = 526,83=C a2 = 553,17=C A2 = 526,83=C I2 = 26,34=C D2 = 0,00=C

Théorème

L’annuité ak de fin de période k est formée d′intérêts et d’amortissement : ak = Ik +Ak ,

les intérêts Ik se calculent sur le capital dû en début de période k : Ik = i ·Dk−1,

le capital dû est diminué de l’amortissement à chaque période : Dk = Dk−1−Ak .


Plan






Taux unique équivalent.

x : taux unique qui donnerait deux annuités de 553,17=C pour unemprunt de 1000=C.


entrées :

1000

553,17 553,17


taux annuel : x = 0,0701 soit 7,01% annuel.





entrées :

1000

553,17 553,17







entrées :

1000

553,17 553,17







entrées :

1000

553,17 553,17







entrées :

1000

553,17 553,17






taux annuel : x taux annuel : xaxe du temps :


entrées :

1000

553,17 553,17








entrées :

1000

553,17 553,17








entrées :

1000

553,17 553,17









entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .







entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .







entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .







entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .

X = 1+ x donne 553,17+553,17X = 1000X 2,







entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .

X = 1+ x donne 553,17+553,17X = 1000X 2,

1000X 2−553,17X −553,17 = 0.







entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .

X = 1+ x donne 553,17+553,17X = 1000X 2,

1000X 2−553,17X −553,17 = 0.

Rappel : aX 2 +bX + c = 0∆ = b2−4ac .Si ∆ > 0 alors

X1 = −b−√

∆2a et X2 = −b+

√∆

2a .







entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .

X = 1+ x donne 553,17+553,17X = 1000X 2,

1000X 2−553,17X −553,17 = 0.


X1 = −b−√

∆2a et X2 = −b+

√∆

2a .







entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .

X = 1+ x donne 553,17+553,17X = 1000X 2,

1000X 2−553,17X −553,17 = 0.


X1 = −b−√

∆2a et X2 = −b+

√∆

2a .







entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .

X = 1+ x donne 553,17+553,17X = 1000X 2,

1000X 2−553,17X −553,17 = 0.


X1 = −b−√

∆2a et X2 = −b+

√∆

2a .







entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .

X = 1+ x donne 553,17+553,17X = 1000X 2,

1000X 2−553,17X −553,17 = 0.


X1 = −b−√

∆2a et X2 = −b+

√∆

2a .∆ = 2518677,0489







entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .

X = 1+ x donne 553,17+553,17X = 1000X 2,

1000X 2−553,17X −553,17 = 0.


X1 = −b−√

∆2a et X2 = −b+

√∆

2a .∆ = 2518677,0489

X1 =−0,5169 donc 1+ x =−0,5169 donc x =−1,5169 (inacceptable).







entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .

X = 1+ x donne 553,17+553,17X = 1000X 2,

1000X 2−553,17X −553,17 = 0.


X1 = −b−√

∆2a et X2 = −b+

√∆

2a .∆ = 2518677,0489








entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .

X = 1+ x donne 553,17+553,17X = 1000X 2,

1000X 2−553,17X −553,17 = 0.


X1 = −b−√

∆2a et X2 = −b+

√∆

2a .∆ = 2518677,0489








entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .

X = 1+ x donne 553,17+553,17X = 1000X 2,

1000X 2−553,17X −553,17 = 0.


X1 = −b−√

∆2a et X2 = −b+

√∆

2a .∆ = 2518677,0489








entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .

X = 1+ x donne 553,17+553,17X = 1000X 2,

1000X 2−553,17X −553,17 = 0.


X1 = −b−√

∆2a et X2 = −b+

√∆

2a .∆ = 2518677,0489


X2 = 1,0701 donc 1+ x = 1,0701 donc x = 0,0701 (acceptable).







entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .

X = 1+ x donne 553,17+553,17X = 1000X 2,

1000X 2−553,17X −553,17 = 0.


X1 = −b−√

∆2a et X2 = −b+

√∆

2a .∆ = 2518677,0489









entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .

X = 1+ x donne 553,17+553,17X = 1000X 2,

1000X 2−553,17X −553,17 = 0.


X1 = −b−√

∆2a et X2 = −b+

√∆

2a .∆ = 2518677,0489









entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .

X = 1+ x donne 553,17+553,17X = 1000X 2,

1000X 2−553,17X −553,17 = 0.


X1 = −b−√

∆2a et X2 = −b+

√∆

2a .∆ = 2518677,0489









entrées :

1000

553,17 553,17



553,17+553,17(1+ x) = 1000(1+ x)2 .

X = 1+ x donne 553,17+553,17X = 1000X 2,

1000X 2−553,17X −553,17 = 0.


X1 = −b−√

∆2a et X2 = −b+

√∆

2a .∆ = 2518677,0489





Plan






Taux actuariel effectif global.

Frais de dossier : 50=C (versé à t = 0), assurance annuelle : 20=C(versé en fin d’année)y : taux annuel effectif assurant l’équivalence entrées-sorties.


entrées :

9501000−50

573,17 573,17

sorties :A une date donnée, les capitaux entrés doivent être équivalents aux capitaux sortis.

TAEG : y = 0,1349 soit 13,49% annuel.





entrées :

9501000−50

573,17 573,17







entrées :

9501000−50

573,17 573,17







entrées :

9501000−50

573,17

553,17+20

573,17







entrées :

9501000−50

573,17 573,17

553,17+20






taux annuel : y taux annuel : yaxe du temps :


entrées :

9501000−50

573,17 573,17








entrées :

9501000−50

573,17 573,17








entrées :

9501000−50

573,17 573,17









entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .







entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .







entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .







entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .

Y = 1+ y donne 573,17+573,17Y = 950Y 2,







entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .

Y = 1+ y donne 573,17+573,17Y = 950Y 2,

950Y 2−573,17Y −573,17 = 0.







entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .

Y = 1+ y donne 573,17+573,17Y = 950Y 2,

950Y 2−573,17Y −573,17 = 0.

Rappel : aY 2 +bY + c = 0∆ = b2−4ac .Si ∆ > 0 alors

Y1 = −b−√

∆2a et Y2 = −b+

√∆

2a .







entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .

Y = 1+ y donne 573,17+573,17Y = 950Y 2,

950Y 2−573,17Y −573,17 = 0.


Y1 = −b−√

∆2a et Y2 = −b+

√∆

2a .







entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .

Y = 1+ y donne 573,17+573,17Y = 950Y 2,

950Y 2−573,17Y −573,17 = 0.


Y1 = −b−√

∆2a et Y2 = −b+

√∆

2a .







entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .

Y = 1+ y donne 573,17+573,17Y = 950Y 2,

950Y 2−573,17Y −573,17 = 0.


Y1 = −b−√

∆2a et Y2 = −b+

√∆

2a .







entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .

Y = 1+ y donne 573,17+573,17Y = 950Y 2,

950Y 2−573,17Y −573,17 = 0.


Y1 = −b−√

∆2a et Y2 = −b+

√∆

2a .∆ = 2506569,8489







entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .

Y = 1+ y donne 573,17+573,17Y = 950Y 2,

950Y 2−573,17Y −573,17 = 0.


Y1 = −b−√

∆2a et Y2 = −b+

√∆

2a .∆ = 2506569,8489

Y1 =−0,5316 donc 1+ y =−0,5316 donc y =−1,5316 (inacceptable).







entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .

Y = 1+ y donne 573,17+573,17Y = 950Y 2,

950Y 2−573,17Y −573,17 = 0.


Y1 = −b−√

∆2a et Y2 = −b+

√∆

2a .∆ = 2506569,8489








entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .

Y = 1+ y donne 573,17+573,17Y = 950Y 2,

950Y 2−573,17Y −573,17 = 0.


Y1 = −b−√

∆2a et Y2 = −b+

√∆

2a .∆ = 2506569,8489








entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .

Y = 1+ y donne 573,17+573,17Y = 950Y 2,

950Y 2−573,17Y −573,17 = 0.


Y1 = −b−√

∆2a et Y2 = −b+

√∆

2a .∆ = 2506569,8489








entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .

Y = 1+ y donne 573,17+573,17Y = 950Y 2,

950Y 2−573,17Y −573,17 = 0.


Y1 = −b−√

∆2a et Y2 = −b+

√∆

2a .∆ = 2506569,8489


Y2 = 1,1349 donc 1+ y = 1,1349 donc y = 0,1349 (acceptable).







entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .

Y = 1+ y donne 573,17+573,17Y = 950Y 2,

950Y 2−573,17Y −573,17 = 0.


Y1 = −b−√

∆2a et Y2 = −b+

√∆

2a .∆ = 2506569,8489









entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .

Y = 1+ y donne 573,17+573,17Y = 950Y 2,

950Y 2−573,17Y −573,17 = 0.


Y1 = −b−√

∆2a et Y2 = −b+

√∆

2a .∆ = 2506569,8489









entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .

Y = 1+ y donne 573,17+573,17Y = 950Y 2,

950Y 2−573,17Y −573,17 = 0.


Y1 = −b−√

∆2a et Y2 = −b+

√∆

2a .∆ = 2506569,8489









entrées :

9501000−50

573,17 573,17



573,17+573,17(1+ y) = 950(1+ y)2 .

Y = 1+ y donne 573,17+573,17Y = 950Y 2,

950Y 2−573,17Y −573,17 = 0.


Y1 = −b−√

∆2a et Y2 = −b+

√∆

2a .∆ = 2506569,8489





Coût du crédit.

Annuités : 2×553,17 = 1106,34=C.

Frais de dossier : 50=C.

Assurance : 2×20 = 40=C.

Le coût du crédit est de

1106,34+50+40 = 1196,34=C.


Coût du crédit.

Annuités : 2×553,17 = 1106,34=C.




1106,34+50+40 = 1196,34=C.


Coût du crédit.

Annuités : 2×553,17 = 1106,34=C.




1106,34+50+40 = 1196,34=C.


Coût du crédit.

Annuités : 2×553,17 = 1106,34=C.




1106,34+50+40 = 1196,34=C.


Plan






Présentation du contexte - Formule des annuités fixes.

Emprunt : D euros (à t = 0).

axe du temps :périodes écoulées : 0 1 2 3 n−1 n

entrées :

sorties :

a = iD1−(1+i)−n





entrées : D

sorties :

a = iD1−(1+i)−n




Annuités simples de fin de période.

Annuités constantes : a euros.


entrées : D

sorties :

a = iD1−(1+i)−n







entrées :

a a a a a

D

sorties :

a = iD1−(1+i)−n






i : taux d’intérêt d’une période de remboursement.

taux périodique : iaxe du temps :

périodes écoulées : 0 1 2 3 n−1 n

entrées :

a a a a a

D

sorties :

a = iD1−(1+i)−n






i : taux d’intérêt d’une période de remboursement.



entrées :

a a a a a

D

sorties :

a = iD1−(1+i)−n


Démonstartion de la formule des annuités fixes.


D = a× 1−(1+i)−n

i



entrées :

a a a a a

D

sorties :




D = a (1+ i)−1 +a (1+ i)−2 + · · ·+a (1+ i)−n

D = a× 1−(1+i)−n

i



entrées :

a a a a a

D

sorties :




D = a (1+ i)−1 +a (1+ i)−2 + · · ·+a (1+ i)−n

D = a[(

(1+ i)−1)1

+(

(1+ i)−1)2

+ · · ·+(

(1+ i)−1)n

]

D = a× 1−(1+i)−n

i



entrées :

a a a a a

D

sorties :



Rappel : q +q2 + · · ·+qn = q× 1−qn

1−q pour q 6= 1.Equation de valeur à t = 0 :

D = a (1+ i)−1 +a (1+ i)−2 + · · ·+a (1+ i)−n

D = a[(

(1+ i)−1)1

+(

(1+ i)−1)2

+ · · ·+(

(1+ i)−1)n

]

D = a× 1−(1+i)−n

i



entrées :

a a a a a

D

sorties :



Rappel : q +q2 + · · ·+qn = q× 1−qn


D = a (1+ i)−1 +a (1+ i)−2 + · · ·+a (1+ i)−n

D = a[(

(1+ i)−1)1

+(

(1+ i)−1)2

+ · · ·+(

(1+ i)−1)n

]D = a (1+ i)−1× 1−((1+i)−1)

n

1−(1+i)−1

D = a× 1−(1+i)−n

i



entrées :

a a a a a

D

sorties :



Rappel : q +q2 + · · ·+qn = q× 1−qn


D = a (1+ i)−1 +a (1+ i)−2 + · · ·+a (1+ i)−n

D = a[(

(1+ i)−1)1

+(

(1+ i)−1)2

+ · · ·+(

(1+ i)−1)n

]D = a (1+ i)−1× 1−((1+i)−1)

n

1−(1+i)−1

D = a× 1(1+i) ×

1−(1+i)−n

1−(1+i)−1

D = a× 1−(1+i)−n

i



entrées :

a a a a a

D

sorties :



Rappel : q +q2 + · · ·+qn = q× 1−qn


D = a (1+ i)−1 +a (1+ i)−2 + · · ·+a (1+ i)−n

D = a[(

(1+ i)−1)1

+(

(1+ i)−1)2

+ · · ·+(

(1+ i)−1)n

]D = a (1+ i)−1× 1−((1+i)−1)

n

1−(1+i)−1

D = a× 1(1+i) ×

1−(1+i)−n

1−(1+i)−1

D = a× 1−(1+i)−n

(1+i)−1

D = a× 1−(1+i)−n

i



entrées :

a a a a a

D

sorties :



Rappel : q +q2 + · · ·+qn = q× 1−qn


D = a (1+ i)−1 +a (1+ i)−2 + · · ·+a (1+ i)−n

D = a[(

(1+ i)−1)1

+(

(1+ i)−1)2

+ · · ·+(

(1+ i)−1)n

]D = a (1+ i)−1× 1−((1+i)−1)

n

1−(1+i)−1

D = a× 1(1+i) ×

1−(1+i)−n

1−(1+i)−1

D = a× 1−(1+i)−n

(1+i)−1

D = a× 1−(1+i)−n

i

D = a× 1−(1+i)−n

i



entrées :

a a a a a

D

sorties :



Rappel : q +q2 + · · ·+qn = q× 1−qn


D = a (1+ i)−1 +a (1+ i)−2 + · · ·+a (1+ i)−n

D = a[(

(1+ i)−1)1

+(

(1+ i)−1)2

+ · · ·+(

(1+ i)−1)n

]D = a (1+ i)−1× 1−((1+i)−1)

n

1−(1+i)−1

D = a× 1(1+i) ×

1−(1+i)−n

1−(1+i)−1

D = a× 1−(1+i)−n

(1+i)−1

D = a× 1−(1+i)−n

i

a = iD1−(1+i)−n



entrées :

a a a a a

D

sorties :


Remarques.

Attention :taux annuel du prêt 6= taux périodique (généralement mensuel).

En France, taux proportionnel : pour les crédits immobiliers etprofessionnels.

En France, taux équivalent : pour les crédits à la consommation.

A l’étranger : taux équivalent généralement utilisé.


Remarques.






Remarques.






Remarques.






Exemple 1.

ExemplePour une maison, Mr Mercers emprunte 100000=C remboursables en 15ans, au taux nominal annuel de 5,1%. Calculer le montant m desmensualités et donner le coût du crédit.

Valeur du taux mensuel i : taux mensuel proportionnel (prêt immobilier).

180×796,01 = 143281,80=C


Exemple 1.



180×796,01 = 143281,80=C


Exemple 1.



180×796,01 = 143281,80=C


Exemple 1.



i = 5,1/10012 = 0,425% soit i = 0,00425.

180×796,01 = 143281,80=C


Exemple 1.



i = 5,1/10012 = 0,425% soit i = 0,00425.

Valeur de la mensualité m : en 15 ans, il y a 15×12 = 180 mois.

180×796,01 = 143281,80=C


Exemple 1.



i = 5,1/10012 = 0,425% soit i = 0,00425.


180×796,01 = 143281,80=C


Exemple 1.



i = 5,1/10012 = 0,425% soit i = 0,00425.


m = 100000×0,004251−(1+0,00425)−180

= 796,01=C.

180×796,01 = 143281,80=C


Exemple 1.



i = 5,1/10012 = 0,425% soit i = 0,00425.


m = 100000×0,004251−(1+0,00425)−180

= 796,01=C.

Coût du crédit :

180×796,01 = 143281,80=C


Exemple 1.



i = 5,1/10012 = 0,425% soit i = 0,00425.


m = 100000×0,004251−(1+0,00425)−180

= 796,01=C.

Coût du crédit :

180×796,01 = 143281,80=C


Exemple 2.

ExemplePour une voiture de luxe, Mr Mercers emprunte 100000=C remboursablesen 15 ans, au taux effectif annuel de 5,1%. Calculer le montant m desmensualités et donner le coût du crédit.

Valeur du taux mensuel i : taux mensuel équivalent (prêt à la consommation).

Commentaire : taux et montant emprunté s’enoncent de la même manière, mais les résultatssont différents.


Exemple 2.





Exemple 2.





Exemple 2.



i = (1+0,051)1/12−1' 0,00415 soit 0,415% (que l’on met en mémoire sans arrondir).



Exemple 2.







Exemple 2.







Exemple 2.





m = 100000×i1−(1+i)−180

= 789,99=C.



Exemple 2.





m = 100000×i1−(1+i)−180

= 789,99=C.

Coût du crédit :



Exemple 2.





m = 100000×i1−(1+i)−180

= 789,99=C.

Coût du crédit :

180×789,99 = 142198,20=C



Exemple 2.





m = 100000×i1−(1+i)−180

= 789,99=C.

Coût du crédit :

180×789,99 = 142198,20=C



Plan






Tableau d’amortissement (dates 1 et 2).

Prêt immobilier de 100000=C au taux nominal annuel 5,1%.

DateCapital dû endébut de mois

Mensualité Amortissement IntérêtCapital dû enfin de mois

1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C







1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

A la date 1 :







1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

A la date 1 :

Dette : D0 = 100000=C.







1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C

×0,00425

D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.







1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.

Annuité : a1 = 796,01=C.







1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C

−

I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.

Annuité : a1 = 796,01=C.








1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

−2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.

Annuité : a1 = 796,01=C.








1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.

Annuité : a1 = 796,01=C.


Reste dû : D1 = 100000−371,01 = 99628,99=C.

A la date 2 :

Reste dû : D2 = 99628,99−372,59 = 99256,40=C.






1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.

Annuité : a1 = 796,01=C.


Reste dû : D1 = 100000−371,01 = 99628,99=C.

A la date 2 :

Dette : D1 = 99628,99=C.

Reste dû : D2 = 99628,99−372,59 = 99256,40=C.






1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C

×0,00425

D2 = 99256,40=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.

Annuité : a1 = 796,01=C.


Reste dû : D1 = 100000−371,01 = 99628,99=C.

A la date 2 :

Dette : D1 = 99628,99=C.

Intérêt : I2 = 99628,99×0,00425 = 423,42=C.

Reste dû : D2 = 99628,99−372,59 = 99256,40=C.






1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.

Annuité : a1 = 796,01=C.


Reste dû : D1 = 100000−371,01 = 99628,99=C.

A la date 2 :

Dette : D1 = 99628,99=C.

Intérêt : I2 = 99628,99×0,00425 = 423,42=C.

Annuité : a2 = 796,01=C.

Reste dû : D2 = 99628,99−372,59 = 99256,40=C.






1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C

−

I2 = 423,42=C D2 = 99256,40=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.

Annuité : a1 = 796,01=C.


Reste dû : D1 = 100000−371,01 = 99628,99=C.

A la date 2 :

Dette : D1 = 99628,99=C.

Intérêt : I2 = 99628,99×0,00425 = 423,42=C.

Annuité : a2 = 796,01=C.


Reste dû : D2 = 99628,99−372,59 = 99256,40=C.






1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

−

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.

Annuité : a1 = 796,01=C.


Reste dû : D1 = 100000−371,01 = 99628,99=C.

A la date 2 :

Dette : D1 = 99628,99=C.

Intérêt : I2 = 99628,99×0,00425 = 423,42=C.

Annuité : a2 = 796,01=C.


Reste dû : D2 = 99628,99−372,59 = 99256,40=C.






3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C

Reste dû : D3 = 99256,40−374,17 = 98882,23=C. Reste dû : D4 = 98882,23−375,76 = 98506,47=C.






3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C

A la date 3 :







3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C

A la date 3 :

Dette : D2 = 99256,40=C.







3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C

×0,00425

D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C

A la date 3 :

Dette : D2 = 99256,40=C.

Intérêt : I3 = 99256,40×0,00425 = 421,84=C.







3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C

A la date 3 :

Dette : D2 = 99256,40=C.

Intérêt : I3 = 99256,40×0,00425 = 421,84=C.

Annuité : a3 = 796,01=C.







3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C

−

I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C

A la date 3 :

Dette : D2 = 99256,40=C.

Intérêt : I3 = 99256,40×0,00425 = 421,84=C.

Annuité : a3 = 796,01=C.








3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

−4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C

A la date 3 :

Dette : D2 = 99256,40=C.

Intérêt : I3 = 99256,40×0,00425 = 421,84=C.

Annuité : a3 = 796,01=C.








3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C

A la date 3 :

Dette : D2 = 99256,40=C.

Intérêt : I3 = 99256,40×0,00425 = 421,84=C.

Annuité : a3 = 796,01=C.


Reste dû : D3 = 99256,40−374,17 = 98882,23=C.

A la date 4 :

Reste dû : D4 = 98882,23−375,76 = 98506,47=C.






3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C

A la date 3 :

Dette : D2 = 99256,40=C.

Intérêt : I3 = 99256,40×0,00425 = 421,84=C.

Annuité : a3 = 796,01=C.


Reste dû : D3 = 99256,40−374,17 = 98882,23=C.

A la date 4 :

Dette : D3 = 98882,23=C.

Reste dû : D4 = 98882,23−375,76 = 98506,47=C.






3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C

×0,00425

D4 = 98506,47=C

A la date 3 :

Dette : D2 = 99256,40=C.

Intérêt : I3 = 99256,40×0,00425 = 421,84=C.

Annuité : a3 = 796,01=C.


Reste dû : D3 = 99256,40−374,17 = 98882,23=C.

A la date 4 :

Dette : D3 = 98882,23=C.

Intérêt : I4 = 98882,23×0,00425 = 420,25=C.

Reste dû : D4 = 98882,23−375,76 = 98506,47=C.






3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C

A la date 3 :

Dette : D2 = 99256,40=C.

Intérêt : I3 = 99256,40×0,00425 = 421,84=C.

Annuité : a3 = 796,01=C.


Reste dû : D3 = 99256,40−374,17 = 98882,23=C.

A la date 4 :

Dette : D3 = 98882,23=C.

Intérêt : I4 = 98882,23×0,00425 = 420,25=C.

Annuité : a4 = 796,01=C.

Reste dû : D4 = 98882,23−375,76 = 98506,47=C.






3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C

−

I4 = 420,25=C D4 = 98506,47=C

A la date 3 :

Dette : D2 = 99256,40=C.

Intérêt : I3 = 99256,40×0,00425 = 421,84=C.

Annuité : a3 = 796,01=C.


Reste dû : D3 = 99256,40−374,17 = 98882,23=C.

A la date 4 :

Dette : D3 = 98882,23=C.

Intérêt : I4 = 98882,23×0,00425 = 420,25=C.

Annuité : a4 = 796,01=C.


Reste dû : D4 = 98882,23−375,76 = 98506,47=C.






3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C

−

A la date 3 :

Dette : D2 = 99256,40=C.

Intérêt : I3 = 99256,40×0,00425 = 421,84=C.

Annuité : a3 = 796,01=C.


Reste dû : D3 = 99256,40−374,17 = 98882,23=C.

A la date 4 :

Dette : D3 = 98882,23=C.

Intérêt : I4 = 98882,23×0,00425 = 420,25=C.

Annuité : a4 = 796,01=C.


Reste dû : D4 = 98882,23−375,76 = 98506,47=C.


Plan






Progression des amortissements.



1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C

Calcul de An/An−1 :





1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C






1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C


A2/A1 ' 1,004258645





1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C


A2/A1 ' 1,004258645

A3/A2 ' 1,004240586





1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C


A2/A1 ' 1,004258645

A3/A2 ' 1,004240586

A4/A3 ' 1,004249405





1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C


A2/A1 ' 1,004258645

A3/A2 ' 1,004240586

A4/A3 ' 1,004249405

An/An−1?= 1,00425.





1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C


A2/A1 ' 1,004258645

A3/A2 ' 1,004240586

A4/A3 ' 1,004249405

An/An−1?= 1,00425.

Expression de A2 en fonction de A1 :





1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C


A2/A1 ' 1,004258645

A3/A2 ' 1,004240586

A4/A3 ' 1,004249405

An/An−1?= 1,00425.


A2 = a− I2 = a− i ×D1 = a− i (D−A1)





1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C


A2/A1 ' 1,004258645

A3/A2 ' 1,004240586

A4/A3 ' 1,004249405

An/An−1?= 1,00425.


A2 = a− I2 = a− i ×D1 = a− i (D−A1)





1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C


A2/A1 ' 1,004258645

A3/A2 ' 1,004240586

A4/A3 ' 1,004249405

An/An−1?= 1,00425.


A2 = a− I2 = a− i ×D1 = a− i (D−A1)





1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C


A2/A1 ' 1,004258645

A3/A2 ' 1,004240586

A4/A3 ' 1,004249405

An/An−1?= 1,00425.


A2 = a− I2 = a− i ×D1 = a− i (D−A1)

A2 = a− i ×D + i ×A1 = A1 + i ×A1





1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C


A2/A1 ' 1,004258645

A3/A2 ' 1,004240586

A4/A3 ' 1,004249405

An/An−1?= 1,00425.


A2 = a− I2 = a− i ×D1 = a− i (D−A1)

A2 = a− i ×D + i ×A1 = A1 + i ×A1





1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C


A2/A1 ' 1,004258645

A3/A2 ' 1,004240586

A4/A3 ' 1,004249405

An/An−1?= 1,00425.


A2 = a− I2 = a− i ×D1 = a− i (D−A1)

A2 = a− i ×D + i ×A1 = A1 + i ×A1

A2 = (1+ i)A1.





1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C×(1+ i)

I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C


A2/A1 ' 1,004258645

A3/A2 ' 1,004240586

A4/A3 ' 1,004249405

An/An−1?= 1,00425.


A2 = a− I2 = a− i ×D1 = a− i (D−A1)

A2 = a− i ×D + i ×A1 = A1 + i ×A1

A2 = (1+ i)A1.





1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C×(1+ i)

I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C×(1+ i)

I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C I4 = 420,25=C D4 = 98506,47=C


A2/A1 ' 1,004258645

A3/A2 ' 1,004240586

A4/A3 ' 1,004249405

An/An−1?= 1,00425.


A2 = a− I2 = a− i ×D1 = a− i (D−A1)

A2 = a− i ×D + i ×A1 = A1 + i ×A1

A2 = (1+ i)A1.





1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C×(1+ i)

I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C×(1+ i)

I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C×(1+ i)

I4 = 420,25=C D4 = 98506,47=C


A2/A1 ' 1,004258645

A3/A2 ' 1,004240586

A4/A3 ' 1,004249405

An/An−1?= 1,00425.


A2 = a− I2 = a− i ×D1 = a− i (D−A1)

A2 = a− i ×D + i ×A1 = A1 + i ×A1

A2 = (1+ i)A1.





1 D0 = 100000,00=C a1 = 796,01=C A1 = 371,01=C I1 = 425=C D1 = 99628,99=C

2 D1 = 99628,99=C a2 = 796,01=C A2 = 372,59=C×(1+ i)

I2 = 423,42=C D2 = 99256,40=C

3 D2 = 99256,40=C a3 = 796,01=C A3 = 374,17=C×(1+ i)

I3 = 421,84=C D3 = 98882,23=C

4 D3 = 98882,23=C a4 = 796,01=C A4 = 375,76=C×(1+ i)

I4 = 420,25=C D4 = 98506,47=C


A2/A1 ' 1,004258645

A3/A2 ' 1,004240586

A4/A3 ' 1,004249405

An/An−1?= 1,00425.


A2 = a− I2 = a− i ×D1 = a− i (D−A1)

A2 = a− i ×D + i ×A1 = A1 + i ×A1

A2 = (1+ i)A1.

ThéorèmeLa suite des amortissements est géométrique de raison (1+ i).

An = A1 (1+ i)n−1 .


Ligne 60 du tableau d’amortissement.



60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :





60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :





60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.





60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.





60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.





60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.





60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.

I60 +A60 = a60 donc I60 = a60−A60 = 796,01−476,49 = 319,52=C.





60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.

I60 +A60 = a60 donc I60 = a60−A60 = 796,01−476,49 = 319,52=C.





60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.

I60 +A60 = a60 donc I60 = a60−A60 = 796,01−476,49 = 319,52=C.





60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C−

D60 = 74704,58=C

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.

I60 +A60 = a60 donc I60 = a60−A60 = 796,01−476,49 = 319,52=C.





60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.

I60 +A60 = a60 donc I60 = a60−A60 = 796,01−476,49 = 319,52=C.

I60 = D59× i et I60 = a60−A60 donc D59× i = a60−A60





60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.

I60 +A60 = a60 donc I60 = a60−A60 = 796,01−476,49 = 319,52=C.






60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.

I60 +A60 = a60 donc I60 = a60−A60 = 796,01−476,49 = 319,52=C.






60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.

I60 +A60 = a60 donc I60 = a60−A60 = 796,01−476,49 = 319,52=C.


D59 = a60−A60i = 796,01−371,01(1+0,00425)59

0,00425 = 75181,07=C





60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.

I60 +A60 = a60 donc I60 = a60−A60 = 796,01−476,49 = 319,52=C.


D59 = a60−A60i = 796,01−371,01(1+0,00425)59

0,00425 = 75181,07=C NE PAS utiliser la valeur approchée de A60.





60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.

I60 +A60 = a60 donc I60 = a60−A60 = 796,01−476,49 = 319,52=C.


D59 = a60−A60i = 796,01−371,01(1+0,00425)59






60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.

I60 +A60 = a60 donc I60 = a60−A60 = 796,01−476,49 = 319,52=C.


D59 = a60−A60i = 796,01−371,01(1+0,00425)59


D60 = D59−A60 = 75181,07−476,49 = 74704,58=C





60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.

I60 +A60 = a60 donc I60 = a60−A60 = 796,01−476,49 = 319,52=C.


D59 = a60−A60i = 796,01−371,01(1+0,00425)59


D60 = D59−A60 = 75181,07−476,49 = 74704,58=C





60 D59 = 75181,07=C a60 = 796,01=C A60 = 476,49=C I60 = 319,52=C D60 = 74704,58=C

−

A la date 60 :

A60 = A1 (1+ i)59 = 371,01(1+0,00425)59 = 476,49=C.

I60 +A60 = a60 donc I60 = a60−A60 = 796,01−476,49 = 319,52=C.


D59 = a60−A60i = 796,01−371,01(1+0,00425)59


D60 = D59−A60 = 75181,07−476,49 = 74704,58=C


Dernière ligne du tableau d’amortissement.



180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :





180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :





180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.





180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.





180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.





180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.





180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.





180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.





180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.





180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C

−

D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.





180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.






180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.






180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.






180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.


D179 = a180−A180i = 796,01−371,01(1+0,00425)179

0,00425 = 793,34=C





180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.


D179 = a180−A180i = 796,01−371,01(1+0,00425)179






180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.


D179 = a180−A180i = 796,01−371,01(1+0,00425)179






180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.


D179 = a180−A180i = 796,01−371,01(1+0,00425)179


D180 = D179−A180 = 793,34−792,64 = 0,70=C





180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.


D179 = a180−A180i = 796,01−371,01(1+0,00425)179


D180 = D179−A180 = 793,34−792,64 = 0,70=C





180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

−

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.


D179 = a180−A180i = 796,01−371,01(1+0,00425)179


D180 = D179−A180 = 793,34−792,64 = 0,70=C





180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.


D179 = a180−A180i = 796,01−371,01(1+0,00425)179


D180 = D179−A180 = 793,34−792,64 = 0,70=C Le prêt n’est pas remboursé !





180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

180 bis D179 = 793,34=C a180 = 796,71=C A180 = 793,34=C I180 = 3,37=C D180 = 0,00=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.


D179 = a180−A180i = 796,01−371,01(1+0,00425)179







180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

180 bis D179 = 793,34=C a180 = 796,71=C A180 = 793,34=C I180 = 3,37=C D180 = 0,00=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.


D179 = a180−A180i = 796,01−371,01(1+0,00425)179







180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

180 bis D179 = 793,34=C a180 = 796,71=C A180 = 793,34=C I180 = 3,37=C D180 = 0,00=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.


D179 = a180−A180i = 796,01−371,01(1+0,00425)179







180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

180 bis D179 = 793,34=C a180 = 796,71=C A180 = 793,34=C I180 = 3,37=C D180 = 0,00=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.


D179 = a180−A180i = 796,01−371,01(1+0,00425)179







180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

180 bis D179 = 793,34=C a180 = 796,71=C+

A180 = 793,34=C I180 = 3,37=C D180 = 0,00=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.


D179 = a180−A180i = 796,01−371,01(1+0,00425)179







180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

180 bis D179 = 793,34=C a180 = 796,71=C A180 = 793,34=C I180 = 3,37=C D180 = 0,00=C

−

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.


D179 = a180−A180i = 796,01−371,01(1+0,00425)179







180 D179 = 793,34=C a180 = 796,01=C A180 = 792,64=C I180 = 3,37=C D180 = 0,70=C

180 bis D179 = 793,34=C a180 = 796,71=C A180 = 793,34=C I180 = 3,37=C D180 = 0,00=C

A la date 180 :

A180 = A1 (1+ i)179 = 371,01(1+0,00425)179 = 792,64=C.

I180 +A180 = a180 donc I180 = a180−A180 = 796,01−792,64 = 3,37=C.


D179 = a180−A180i = 796,01−371,01(1+0,00425)179


D180 = D179−A180 = 793,34−792,64 = 0,70=C Le prêt est remboursé !


Plan






Frais de dossier et d’assurance.

Prêt immobilier : 100000 euros (à t = 0).

axe du temps :périodes écoulées : 0 1 2 3 179 180

entrées :

854,01

100000

sorties :

Déterminer x pour que les capitaux entrés et sortis soient équivalents.




Frais de dossier : 400 euros (à t = 0).


entrées :

854,01

100000

400sorties :






Assurance : 58 euros à chaque mensualité.


entrées :

854,01

100000

400sorties :








entrées :

854,01

796,01+58

854,01 854,01 854,01 854,01

100000

400sorties :







x : taux d’intérêt mensuel.

taux mensuel : xaxe du temps :

périodes écoulées : 0 1 2 3 179 180

entrées :

854,01 854,01 854,01 854,01 854,01

100000

400sorties :










entrées :

854,01 854,01 854,01 854,01 854,01

99600

sorties :










entrées :

854,01 854,01 854,01 854,01 854,01

99600

sorties :



Equation vérifiée par x .

Utilisation de la formule des annuités constantes.



entrées :

854,01 854,01 854,01 854,01 854,01

99600

sorties :



Utilisation de la formule des annuités constantes.Utilisation de la formule des annuités constantes.



entrées :

854,01 854,01 854,01 854,01 854,01

99600

sorties :




La situation correspond à un emprunt de 99600=C remboursé en 180 mensualités de 854,01=C.



entrées :

854,01 854,01 854,01 854,01 854,01

99600

sorties :





854,01 = 99600x1−(1+x)−180



entrées :

854,01 854,01 854,01 854,01 854,01

99600

sorties :





854,01 = 99600x1−(1+x)−180

Interprétation de l’équation.



entrées :

854,01 854,01 854,01 854,01 854,01

99600

sorties :





854,01 = 99600x1−(1+x)−180

Interprétation de l’équation.On pose f (x) = 99600x

1−(1+x)−180.



entrées :

854,01 854,01 854,01 854,01 854,01

99600

sorties :





854,01 = 99600x1−(1+x)−180


1−(1+x)−180.

f (x) représente la mensualité à payer pour un taux mensuel x .



entrées :

854,01 854,01 854,01 854,01 854,01

99600

sorties :





854,01 = 99600x1−(1+x)−180


1−(1+x)−180.

f (x) représente la mensualité à payer pour un taux mensuel x .

On cherche la valeur de x pour laquelle f (x) (mensualité à payer) vaut 854,01.



entrées :

854,01 854,01 854,01 854,01 854,01

99600

sorties :


Encadrement de x .

Justifier que x est compris entre 0,5% et 0,6%.

x est compris entre 0,5% et 0,6%.


Encadrement de x .




Encadrement de x .


f (0,005) = 840,48 : au taux mensuel 0,5% la mensualité est de 840,48=C.



Encadrement de x .





Encadrement de x .






Encadrement de x .






Encadrement de x .




taux mensuel

mensualitéReprésentation graphique

de la fonction f



Encadrement de x .




840,48

0,005 taux mensuel


de la fonction f



Encadrement de x .




906,41

840,48

0,005 0,006 taux mensuel


de la fonction f



Encadrement de x .




906,41

840,48



de la fonction f



Encadrement de x .




906,41

854,01

840,48



de la fonction f



Encadrement de x .




906,41

854,01

840,48



de la fonction f



Encadrement de x .




906,41

854,01

840,48

0,005 x 0,006 taux mensuel


de la fonction f



Encadrement de x .




906,41

854,01

840,48



de la fonction f



Valeur approchée de x par interpolation linéaire.

Schéma d’interpolation linéaire.

906,41

854,01

840,48



de la fonction f

x ' 0,0052052 soit 0,52022% mensuel.




906,41

854,01

840,48



de la fonction f

x ' 0,0052052 soit 0,52022% mensuel.




840,48 906,41

+65,93

906,41

854,01

840,48

0,005 x?? 0,006 taux mensuel


de la fonction f

x ' 0,0052052 soit 0,52022% mensuel.




mensualité :

taux mensuel :

840,48 906,41

+65,93

906,41

854,01

840,48



de la fonction f

x ' 0,0052052 soit 0,52022% mensuel.




mensualité :

taux mensuel :

840,48

0,005

906,41

+65,93

906,41

854,01

840,48



de la fonction f

x ' 0,0052052 soit 0,52022% mensuel.




mensualité :

taux mensuel :

840,48

0,005

906,41

0,006

+65,93

906,41

854,01

840,48



de la fonction f

x ' 0,0052052 soit 0,52022% mensuel.




mensualité :

taux mensuel :

840,48

0,005

906,41

0,006

854,01

??

+65,93

906,41

854,01

840,48



de la fonction f

x ' 0,0052052 soit 0,52022% mensuel.




mensualité :

taux mensuel :

840,48

0,005

906,41

0,006

854,01

??

+0,001

+65,93

906,41

854,01

840,48



de la fonction f

x ' 0,0052052 soit 0,52022% mensuel.




mensualité :

taux mensuel :

840,48

0,005

906,41

0,006

854,01

??

+0,001

+65,93

906,41

854,01

840,48



de la fonction f

x ' 0,0052052 soit 0,52022% mensuel.




mensualité :

taux mensuel :

840,48

0,005

906,41

0,006

854,01

??

+0,001

+65,93

+v

906,41

854,01

840,48



de la fonction f

x ' 0,0052052 soit 0,52022% mensuel.




mensualité :

taux mensuel :

840,48

0,005

906,41

0,006

854,01

??

+0,001

+65,93

+v

+13,53

906,41

854,01

840,48



de la fonction f

x ' 0,0052052 soit 0,52022% mensuel.




mensualité :

taux mensuel :

840,48

0,005

906,41

0,006

854,01

??

+0,001

+65,93

+v

+13,53

v13,53 = 0,001

65,93 donc v = 13,53×0,00165,93

906,41

854,01

840,48



de la fonction f

x ' 0,0052052 soit 0,52022% mensuel.




mensualité :

taux mensuel :

840,48

0,005

906,41

0,006

854,01

??

+0,001

+65,93

+v

+13,53

v13,53 = 0,001

65,93 donc v = 13,53×0,00165,93

906,41

854,01

840,48



de la fonction f

x ' 0,0052052 soit 0,52022% mensuel.




mensualité :

taux mensuel :

840,48

0,005

906,41

0,006

854,01

??

+0,001

+65,93

+v

+13,53

v13,53 = 0,001

65,93 donc v = 13,53×0,00165,93

?? = 0,005+ v = 0,005+ 13,53×0,00165,93 ' 0,0052052.

906,41

854,01

840,48



de la fonction f

x ' 0,0052052 soit 0,52022% mensuel.




mensualité :

taux mensuel :

840,48

0,005

906,41

0,006

854,01

??

+0,001

+65,93

+v

+13,53

v13,53 = 0,001

65,93 donc v = 13,53×0,00165,93

?? = 0,005+ v = 0,005+ 13,53×0,00165,93 ' 0,0052052.

906,41

854,01

840,48



de la fonction f

x ' 0,0052052 soit 0,52022% mensuel.




mensualité :

taux mensuel :

840,48

0,005

906,41

0,006

854,01

??

+0,001

+65,93

+v

+13,53

v13,53 = 0,001

65,93 donc v = 13,53×0,00165,93

?? = 0,005+ v = 0,005+ 13,53×0,00165,93 ' 0,0052052.

906,41

854,01

840,48



de la fonction f

x ' 0,0052052 soit 0,52052% mensuel.


TEG et TAEG.

TEG : taux effectif global.

TAEG = (1+ x)12−1' (1+0,0052052)12−1' 0,0643 soit un TAEG de 6,43% annuel.


TEG et TAEG.


Le taux effectif global est le taux annuel proportionnel :



TEG et TAEG.



TEG = 12x ' 12×0,0052052' 0,0625 soit un TEG de 6,25% annuel.



TEG et TAEG.




TAEG : taux actuariel effectif global.



TEG et TAEG.





Le taux actuariel effectif global est le taux annuel équivalent :



TEG et TAEG.





Le taux actuariel effectif global est le taux annuel équivalent :



Plan






Présentation du contexte - Formule des amortissements.

On emprunte D euros (à t = 0).

On rembourse en n périodes.

Les annuités sont versées en fin de période.

Dans chaque annuité, l’amortissement Ak est le même.

A = Dn .

ExempleOn emprunte, pour une maison, 100000=C remboursables en 15 ans àamortissement constant au taux nominal annuel de 5,1%. Calculer lemontant A des amortissements.

A = 100000180 = 555,56=C.







A = Dn .


A = 100000180 = 555,56=C.







A = Dn .


A = 100000180 = 555,56=C.







A = Dn .


A = 100000180 = 555,56=C.







A = Dn .


A = 100000180 = 555,56=C.







A = Dn .


A = 100000180 = 555,56=C.







A = Dn .


A = 100000180 = 555,56=C.


Plan







Prêt immobilier de 100000=C au taux nominal annuel 5,1%(amortissement constant).



1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C







1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

A la date 1 :







1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

A la date 1 :

Dette : D0 = 100000=C.







1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C

×0,00425

D1 = 99444,44=C

2 D1 = 99444,44=C a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.







1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.

Amortissement : A1 = 555,56=C.







1 D0 = 100000,00=C a1 = 980,56=C+

A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.


Annuité : a1 = 425+555,56 = 980,56=C.







1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

−2 D1 = 99444,44=C a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.


Annuité : a1 = 425+555,56 = 980,56=C.







1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.


Annuité : a1 = 425+555,56 = 980,56=C.

Reste dû : D1 = 100000−555,56 = 99444,44=C.

A la date 2 :

Reste dû : D2 = 99444,44−555,56 = 98888,88=C.






1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.


Annuité : a1 = 425+555,56 = 980,56=C.

Reste dû : D1 = 100000−555,56 = 99444,44=C.

A la date 2 :

Dette : D1 = 99444,44=C.

Reste dû : D2 = 99444,44−555,56 = 98888,88=C.






1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C

×0,00425

D2 = 98888,88=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.


Annuité : a1 = 425+555,56 = 980,56=C.

Reste dû : D1 = 100000−555,56 = 99444,44=C.

A la date 2 :

Dette : D1 = 99444,44=C.

Intérêt : I2 = 99444,44×0,00425 = 422,64=C.

Reste dû : D2 = 99444,44−555,56 = 98888,88=C.






1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.


Annuité : a1 = 425+555,56 = 980,56=C.

Reste dû : D1 = 100000−555,56 = 99444,44=C.

A la date 2 :

Dette : D1 = 99444,44=C.

Intérêt : I2 = 99444,44×0,00425 = 422,64=C.


Reste dû : D2 = 99444,44−555,56 = 98888,88=C.






1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C a2 = 978,20=C+

A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.


Annuité : a1 = 425+555,56 = 980,56=C.

Reste dû : D1 = 100000−555,56 = 99444,44=C.

A la date 2 :

Dette : D1 = 99444,44=C.

Intérêt : I2 = 99444,44×0,00425 = 422,64=C.


Annuité : a2 = 422,64+555,56 = 978,20=C.

Reste dû : D2 = 99444,44−555,56 = 98888,88=C.






1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

−

A la date 1 :

Dette : D0 = 100000=C.

Intérêt : I1 = 100000×0,00425 = 425=C.


Annuité : a1 = 425+555,56 = 980,56=C.

Reste dû : D1 = 100000−555,56 = 99444,44=C.

A la date 2 :

Dette : D1 = 99444,44=C.

Intérêt : I2 = 99444,44×0,00425 = 422,64=C.


Annuité : a2 = 422,64+555,56 = 978,20=C.

Reste dû : D2 = 99444,44−555,56 = 98888,88=C.






3 D2 = 98888,88=C a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C







3 D2 = 98888,88=C a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C

A la date 3 :







3 D2 = 98888,88=C a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C

A la date 3 :

Dette : D2 = 98888,88=C.







3 D2 = 98888,88=C a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C

×0,00425

D3 = 98333,32=C

4 D3 = 98333,32=C a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C

A la date 3 :

Dette : D2 = 98888,88=C.

Intérêt : I3 = 98888,88×0,00425 = 420,28=C.







3 D2 = 98888,88=C a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C

A la date 3 :

Dette : D2 = 98888,88=C.

Intérêt : I3 = 98888,88×0,00425 = 420,28=C.








3 D2 = 98888,88=C a3 = 975,84=C+

A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C

A la date 3 :

Dette : D2 = 98888,88=C.

Intérêt : I3 = 98888,88×0,00425 = 420,28=C.


Annuité : a3 = 420,28+555,56 = 975,84=C.







3 D2 = 98888,88=C a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

−4 D3 = 98333,32=C a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C

A la date 3 :

Dette : D2 = 98888,88=C.

Intérêt : I3 = 98888,88×0,00425 = 420,28=C.


Annuité : a3 = 420,28+555,56 = 975,84=C.







3 D2 = 98888,88=C a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C

A la date 3 :

Dette : D2 = 98888,88=C.

Intérêt : I3 = 98888,88×0,00425 = 420,28=C.


Annuité : a3 = 420,28+555,56 = 975,84=C.

Reste dû : D3 = 98888,88−555,56 = 98333,32=C.

A la date 4 :

Reste dû : D4 = 98333,32−555,56 = 97777,76=C.






3 D2 = 98888,88=C a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C

A la date 3 :

Dette : D2 = 98888,88=C.

Intérêt : I3 = 98888,88×0,00425 = 420,28=C.


Annuité : a3 = 420,28+555,56 = 975,84=C.

Reste dû : D3 = 98888,88−555,56 = 98333,32=C.

A la date 4 :

Dette : D3 = 98333,32=C.

Reste dû : D4 = 98333,32−555,56 = 97777,76=C.






3 D2 = 98888,88=C a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C

×0,00425

D4 = 97777,76=C

A la date 3 :

Dette : D2 = 98888,88=C.

Intérêt : I3 = 98888,88×0,00425 = 420,28=C.


Annuité : a3 = 420,28+555,56 = 975,84=C.

Reste dû : D3 = 98888,88−555,56 = 98333,32=C.

A la date 4 :

Dette : D3 = 98333,32=C.

Intérêt : I4 = 98333,32×0,00425 = 417,92=C.

Reste dû : D4 = 98333,32−555,56 = 97777,76=C.






3 D2 = 98888,88=C a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C

A la date 3 :

Dette : D2 = 98888,88=C.

Intérêt : I3 = 98888,88×0,00425 = 420,28=C.


Annuité : a3 = 420,28+555,56 = 975,84=C.

Reste dû : D3 = 98888,88−555,56 = 98333,32=C.

A la date 4 :

Dette : D3 = 98333,32=C.

Intérêt : I4 = 98333,32×0,00425 = 417,92=C.


Reste dû : D4 = 98333,32−555,56 = 97777,76=C.






3 D2 = 98888,88=C a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C a4 = 973,48=C+

A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C

A la date 3 :

Dette : D2 = 98888,88=C.

Intérêt : I3 = 98888,88×0,00425 = 420,28=C.


Annuité : a3 = 420,28+555,56 = 975,84=C.

Reste dû : D3 = 98888,88−555,56 = 98333,32=C.

A la date 4 :

Dette : D3 = 98333,32=C.

Intérêt : I4 = 98333,32×0,00425 = 417,92=C.


Annuité : a4 = 417,92+555,56 = 973,48=C.

Reste dû : D4 = 98333,32−555,56 = 97777,76=C.






3 D2 = 98888,88=C a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C

−

A la date 3 :

Dette : D2 = 98888,88=C.

Intérêt : I3 = 98888,88×0,00425 = 420,28=C.


Annuité : a3 = 420,28+555,56 = 975,84=C.

Reste dû : D3 = 98888,88−555,56 = 98333,32=C.

A la date 4 :

Dette : D3 = 98333,32=C.

Intérêt : I4 = 98333,32×0,00425 = 417,92=C.


Annuité : a4 = 417,92+555,56 = 973,48=C.

Reste dû : D4 = 98333,32−555,56 = 97777,76=C.


Progression du capital dû et des intérêts.



1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

3 D2 = 98888,88=C a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C

Progression des Dn :





1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

3 D2 = 98888,88=C a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C






1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C−555,56

a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

3 D2 = 98888,88=C a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C


D1 = D0−555,56.





1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C−555,56

a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

3 D2 = 98888,88=C−555,56

a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C


D1 = D0−555,56.

D2 = D1−555,56.





1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C−555,56

a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

3 D2 = 98888,88=C−555,56

a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C−555,56

a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C


D1 = D0−555,56.

D2 = D1−555,56.

D3 = D2−555,56.





1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C−555,56

a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

3 D2 = 98888,88=C−555,56

a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C−555,56

a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C


D1 = D0−555,56.

D2 = D1−555,56.

D3 = D2−555,56.

Dn = Dn−1−555,56.





1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C−555,56

a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

3 D2 = 98888,88=C−555,56

a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C−555,56

a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C


D1 = D0−555,56.

D2 = D1−555,56.

D3 = D2−555,56.

Dn = Dn−1−555,56.

Expression de I2 en fonction de I1 :





1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C−555,56

a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

3 D2 = 98888,88=C−555,56

a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C−555,56

a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C


D1 = D0−555,56.

D2 = D1−555,56.

D3 = D2−555,56.

Dn = Dn−1−555,56.


I2 = D1× i = (D0−A)× i = D0× i −A× i





1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C−555,56

a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

3 D2 = 98888,88=C−555,56

a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C−555,56

a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C


D1 = D0−555,56.

D2 = D1−555,56.

D3 = D2−555,56.

Dn = Dn−1−555,56.


I2 = D1× i = (D0−A)× i = D0× i −A× i





1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C−555,56

a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

3 D2 = 98888,88=C−555,56

a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C−555,56

a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C


D1 = D0−555,56.

D2 = D1−555,56.

D3 = D2−555,56.

Dn = Dn−1−555,56.


I2 = D1× i = (D0−A)× i = D0× i −A× i





1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C−555,56

a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

3 D2 = 98888,88=C−555,56

a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C−555,56

a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C


D1 = D0−555,56.

D2 = D1−555,56.

D3 = D2−555,56.

Dn = Dn−1−555,56.


I2 = D1× i = (D0−A)× i = D0× i −A× i

I2 = I1−A× i = I1−555,56×0,00425





1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C−555,56

a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

3 D2 = 98888,88=C−555,56

a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C−555,56

a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C


D1 = D0−555,56.

D2 = D1−555,56.

D3 = D2−555,56.

Dn = Dn−1−555,56.


I2 = D1× i = (D0−A)× i = D0× i −A× i

I2 = I1−A× i = I1−555,56×0,00425





1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C−555,56

a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C D2 = 98888,88=C

3 D2 = 98888,88=C−555,56

a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C−555,56

a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C


D1 = D0−555,56.

D2 = D1−555,56.

D3 = D2−555,56.

Dn = Dn−1−555,56.


I2 = D1× i = (D0−A)× i = D0× i −A× i

I2 = I1−A× i = I1−555,56×0,00425

I2 = I1−2,36.





1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C−555,56

a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C−2,36

D2 = 98888,88=C

3 D2 = 98888,88=C−555,56

a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C D3 = 98333,32=C

4 D3 = 98333,32=C−555,56

a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C


D1 = D0−555,56.

D2 = D1−555,56.

D3 = D2−555,56.

Dn = Dn−1−555,56.


I2 = D1× i = (D0−A)× i = D0× i −A× i

I2 = I1−A× i = I1−555,56×0,00425

I2 = I1−2,36.





1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C−555,56

a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C−2,36

D2 = 98888,88=C

3 D2 = 98888,88=C−555,56

a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C−2,36

D3 = 98333,32=C

4 D3 = 98333,32=C−555,56

a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C D4 = 97777,76=C


D1 = D0−555,56.

D2 = D1−555,56.

D3 = D2−555,56.

Dn = Dn−1−555,56.


I2 = D1× i = (D0−A)× i = D0× i −A× i

I2 = I1−A× i = I1−555,56×0,00425

I2 = I1−2,36.





1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C−555,56

a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C−2,36

D2 = 98888,88=C

3 D2 = 98888,88=C−555,56

a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C−2,36

D3 = 98333,32=C

4 D3 = 98333,32=C−555,56

a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C−2,36

D4 = 97777,76=C


D1 = D0−555,56.

D2 = D1−555,56.

D3 = D2−555,56.

Dn = Dn−1−555,56.


I2 = D1× i = (D0−A)× i = D0× i −A× i

I2 = I1−A× i = I1−555,56×0,00425

I2 = I1−2,36.





1 D0 = 100000,00=C a1 = 980,56=C A1 = 555,56=C I1 = 425=C D1 = 99444,44=C

2 D1 = 99444,44=C−555,56

a2 = 978,20=C A2 = 555,56=C I2 = 422,64=C−2,36

D2 = 98888,88=C

3 D2 = 98888,88=C−555,56

a3 = 975,84=C A3 = 555,56=C I3 = 420,28=C−2,36

D3 = 98333,32=C

4 D3 = 98333,32=C−555,56

a4 = 973,48=C A4 = 555,56=C I4 = 417,92=C−2,36

D4 = 97777,76=C


D1 = D0−555,56.

D2 = D1−555,56.

D3 = D2−555,56.

Dn = Dn−1−555,56.


I2 = D1× i = (D0−A)× i = D0× i −A× i

I2 = I1−A× i = I1−555,56×0,00425

I2 = I1−2,36.

ThéorèmeLa suite des capitaux dûs (resp. des intérêts) est arithmétique de raison−A (resp. −A× i ).

Dn = D0−n×A.

In = D0× i − (n−1)× i ×A.





60 D59 = 67221,96=C a60 = 841,26=C A60 = 555,56=C I60 = 285,69=C D60 = 66666,40=C

A la date 60 :





60 D59 = 67221,96=C a60 = 841,26=C A60 = 555,56=C I60 = 285,69=C D60 = 66666,40=C

A la date 60 :





60 D59 = 67221,96=C a60 = 841,26=C A60 = 555,56=C I60 = 285,69=C D60 = 66666,40=C

A la date 60 :

D59 = D0−59×A = 100000−59×555,56 = 67221,96=C.





60 D59 = 67221,96=C a60 = 841,26=C A60 = 555,56=C I60 = 285,69=C D60 = 66666,40=C

A la date 60 :

D59 = D0−59×A = 100000−59×555,56 = 67221,96=C.





60 D59 = 67221,96=C a60 = 841,26=C A60 = 555,56=C I60 = 285,69=C D60 = 66666,40=C

A la date 60 :

D59 = D0−59×A = 100000−59×555,56 = 67221,96=C.





60 D59 = 67221,96=C a60 = 841,26=C A60 = 555,56=C I60 = 285,69=C D60 = 66666,40=C

A la date 60 :

D59 = D0−59×A = 100000−59×555,56 = 67221,96=C.

I60 = D59× i = 67221,96×0,00425 = 285,69.





60 D59 = 67221,96=C a60 = 841,26=C A60 = 555,56=C I60 = 285,69=C D60 = 66666,40=C

A la date 60 :

D59 = D0−59×A = 100000−59×555,56 = 67221,96=C.

I60 = D59× i = 67221,96×0,00425 = 285,69.





60 D59 = 67221,96=C a60 = 841,26=C A60 = 555,56=C I60 = 285,69=C

×0,00425

D60 = 66666,40=C

A la date 60 :

D59 = D0−59×A = 100000−59×555,56 = 67221,96=C.

I60 = D59× i = 67221,96×0,00425 = 285,69.





60 D59 = 67221,96=C a60 = 841,26=C A60 = 555,56=C I60 = 285,69=C D60 = 66666,40=C

A la date 60 :

D59 = D0−59×A = 100000−59×555,56 = 67221,96=C.

I60 = D59× i = 67221,96×0,00425 = 285,69.





60 D59 = 67221,96=C a60 = 841,26=C A60 = 555,56=C I60 = 285,69=C D60 = 66666,40=C

A la date 60 :

D59 = D0−59×A = 100000−59×555,56 = 67221,96=C.

I60 = D59× i = 67221,96×0,00425 = 285,69.

a60 = I60 +A = 285,69+555,56 = 841,26.





60 D59 = 67221,96=C a60 = 841,26=C A60 = 555,56=C I60 = 285,69=C D60 = 66666,40=C

A la date 60 :

D59 = D0−59×A = 100000−59×555,56 = 67221,96=C.

I60 = D59× i = 67221,96×0,00425 = 285,69.

a60 = I60 +A = 285,69+555,56 = 841,26.





60 D59 = 67221,96=C a60 = 841,26=C+

A60 = 555,56=C I60 = 285,69=C D60 = 66666,40=C

A la date 60 :

D59 = D0−59×A = 100000−59×555,56 = 67221,96=C.

I60 = D59× i = 67221,96×0,00425 = 285,69.

a60 = I60 +A = 285,69+555,56 = 841,26.





60 D59 = 67221,96=C a60 = 841,26=C A60 = 555,56=C I60 = 285,69=C D60 = 66666,40=C

A la date 60 :

D59 = D0−59×A = 100000−59×555,56 = 67221,96=C.

I60 = D59× i = 67221,96×0,00425 = 285,69.

a60 = I60 +A = 285,69+555,56 = 841,26.

D60 = D59−A = 67221,96−555,56 = 66666,40=C.





60 D59 = 67221,96=C a60 = 841,26=C A60 = 555,56=C I60 = 285,69=C D60 = 66666,40=C

A la date 60 :

D59 = D0−59×A = 100000−59×555,56 = 67221,96=C.

I60 = D59× i = 67221,96×0,00425 = 285,69.

a60 = I60 +A = 285,69+555,56 = 841,26.

D60 = D59−A = 67221,96−555,56 = 66666,40=C.





60 D59 = 67221,96=C a60 = 841,26=C A60 = 555,56=C I60 = 285,69=C D60 = 66666,40=C

−

A la date 60 :

D59 = D0−59×A = 100000−59×555,56 = 67221,96=C.

I60 = D59× i = 67221,96×0,00425 = 285,69.

a60 = I60 +A = 285,69+555,56 = 841,26.

D60 = D59−A = 67221,96−555,56 = 66666,40=C.





180 D179 = 554,76=C a180 = 557,12=C A180 = 554,76=C I180 = 2,36=C D180 = 0,00=C

A la date 60 :





180 D179 = 554,76=C a180 = 557,12=C A180 = 554,76=C I180 = 2,36=C D180 = 0,00=C

A la date 180 :





180 D179 = 554,76=C a180 = 557,12=C A180 = 554,76=C I180 = 2,36=C D180 = 0,00=C

A la date 180 :

D179 = D0−179×A = 100000−179×555,56 = 554,76=C.





180 D179 = 554,76=C a180 = 557,12=C A180 = 554,76=C I180 = 2,36=C D180 = 0,00=C

A la date 180 :

D179 = D0−179×A = 100000−179×555,56 = 554,76=C.





180 D179 = 554,76=C a180 = 557,12=C A180 = 554,76=C I180 = 2,36=C D180 = 0,00=C

A la date 180 :

D179 = D0−179×A = 100000−179×555,56 = 554,76=C.





180 D179 = 554,76=C a180 = 557,12=C A180 = 554,76=C I180 = 2,36=C D180 = 0,00=C

A la date 180 :

D179 = D0−179×A = 100000−179×555,56 = 554,76=C.

I180 = D179× i = 554,76×0,00425 = 2,36.





180 D179 = 554,76=C a180 = 557,12=C A180 = 554,76=C I180 = 2,36=C D180 = 0,00=C

A la date 180 :

D179 = D0−179×A = 100000−179×555,56 = 554,76=C.

I180 = D179× i = 554,76×0,00425 = 2,36.





180 D179 = 554,76=C a180 = 557,12=C A180 = 554,76=C I180 = 2,36=C

×0,00425

D180 = 0,00=C

A la date 180 :

D179 = D0−179×A = 100000−179×555,56 = 554,76=C.

I180 = D179× i = 554,76×0,00425 = 2,36.





180 D179 = 554,76=C a180 = 557,12=C A180 = 554,76=C I180 = 2,36=C D180 = 0,00=C

A la date 180 :

D179 = D0−179×A = 100000−179×555,56 = 554,76=C.

I180 = D179× i = 554,76×0,00425 = 2,36.





180 D179 = 554,76=C a180 = 557,12=C A180 = 554,76=C I180 = 2,36=C D180 = 0,00=C

A la date 180 :

D179 = D0−179×A = 100000−179×555,56 = 554,76=C.

I180 = D179× i = 554,76×0,00425 = 2,36.

a180 = I180 +A = 2,36+554,76 = 557,12.





180 D179 = 554,76=C a180 = 557,12=C A180 = 554,76=C I180 = 2,36=C D180 = 0,00=C

A la date 180 :

D179 = D0−179×A = 100000−179×555,56 = 554,76=C.

I180 = D179× i = 554,76×0,00425 = 2,36.

a180 = I180 +A = 2,36+554,76 = 557,12.





180 D179 = 554,76=C a180 = 557,12=C+

A180 = 554,76=C I180 = 2,36=C D180 = 0,00=C

A la date 180 :

D179 = D0−179×A = 100000−179×555,56 = 554,76=C.

I180 = D179× i = 554,76×0,00425 = 2,36.

a180 = I180 +A = 2,36+554,76 = 557,12.





180 D179 = 554,76=C a180 = 557,12=C A180 = 554,76=C I180 = 2,36=C D180 = 0,00=C

A la date 180 :

D179 = D0−179×A = 100000−179×555,56 = 554,76=C.

I180 = D179× i = 554,76×0,00425 = 2,36.

a180 = I180 +A = 2,36+554,76 = 557,12.

D180 = D179−A180 = 554,76−554,76 = 0,00=C.





180 D179 = 554,76=C a180 = 557,12=C A180 = 554,76=C I180 = 2,36=C D180 = 0,00=C

A la date 180 :

D179 = D0−179×A = 100000−179×555,56 = 554,76=C.

I180 = D179× i = 554,76×0,00425 = 2,36.

a180 = I180 +A = 2,36+554,76 = 557,12.

D180 = D179−A180 = 554,76−554,76 = 0,00=C.





180 D179 = 554,76=C a180 = 557,12=C A180 = 554,76=C I180 = 2,36=C D180 = 0,00=C

−

A la date 180 :

D179 = D0−179×A = 100000−179×555,56 = 554,76=C.

I180 = D179× i = 554,76×0,00425 = 2,36.

a180 = I180 +A = 2,36+554,76 = 557,12.

D180 = D179−A180 = 554,76−554,76 = 0,00=C.





180 D179 = 554,76=C a180 = 557,12=C A180 = 554,76=C I180 = 2,36=C D180 = 0,00=C

A la date 180 :

D179 = D0−179×A = 100000−179×555,56 = 554,76=C.

I180 = D179× i = 554,76×0,00425 = 2,36.

a180 = I180 +A = 2,36+554,76 = 557,12.

D180 = D179−A180 = 554,76−554,76 = 0,00=C. Le prêt est remboursé !





180 D179 = 554,76=C a180 = 557,12=C A180 = 554,76=C I180 = 2,36=C D180 = 0,00=C

A la date 180 :

D179 = D0−179×A = 100000−179×555,56 = 554,76=C.

I180 = D179× i = 554,76×0,00425 = 2,36.

a180 = I180 +A = 2,36+554,76 = 557,12.

D180 = D179−A180 = 554,76−554,76 = 0,00=C. Le prêt est remboursé !

Remarque : le dernier amortissement n’a pas la même valeur que les autres.

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