dh-Materialien
Einführung in Maple    
Anwendungen
 

Abzahlung eines Darlehens

B(0) Darlehensbetrag (Schulden zur Zeit t = 0)
t    Zeit in Monaten nach Aufnahme des Darlehens
a  monatliche Zahlrate ( a = Z + b)
p % Zinssatz p.a. 
Z monatliche Zinsen (Z = k*B(t) mit k = p/1200)
b monatlicher Abtrag
tD gesamte Darlehenszeit
B(t) Schulden zur Zeit t

> restart; with(plots):
  f:= t -> B(t):
  diff(f(t), t) = k*f(t) - a;

d/dt(B(t)) = k*B(t) - a

Lösung dieser Differentialgleichung:

> dsolve(%, B(t));

B(t) = a/k + e^(k*t)*_C1

 Zur Zeit t = 0 wird der gesamte Darlehensbetrag geschuldet:

> lsg:= %:
  subs({B(t)= B[0], t= 0}, %);

B[0] = a/k + e^0 * _C1

Berechnung der Konstanten _C1:

> _C1:= (solve(%, _C1));

_C1 = - (-B[0]*k + a)/k

Die zur Zeit t noch bestehenden Schulden B(t):

> B(t):= simplify(rhs(lsg));

Die monatliche Zahlrate a in Abhängigkeit von der gewünschten Darlehenszeit tD:

> solve(B(t) = 0, a):
  a:= subs(t= t[D], %);

a = e^(k*t[D])*B[0]*k/(-1 + e^(k*t[D]))

Beispiel: Darlehen 10.000 EUR; Zinssatz 7%; zurückzuzahlen in insgesamt 8 Jahren

> B[0]:= 10000:  # Darlehensbetrag
  p:= 7:         # Zinssatz p.a.
  k:= evalf(p/1200):
  t[D]:= 8*12-1: # Darlehenszeit in Monaten
  a:= round(a);  # monatlicher Zahlbetrag

a:= 137

Diskrete Rechnung in monatlichen Schritten:

> t:= 0: Z:= 0: b:= 0:
  b_ges:= 0: Z_ges:= 0:
  S:= B[0]:
  printf(" t S b b_ges Z Z_ges\n");
  printf(" %3.0f %9.2f %7.2f %9.2f %7.2f %9.2f\n",
            t,   S,   b,   b_ges,   Z,   Z_ges);
  P[1]:= [0, 0]:

  while (S > 0) do
    t:= t+1:     # Zeit in Monaten
    Z:= k*S:     # Zinsen im jeweiligen Monat
    b:= a - Z:   # Abtrag im jeweiligen Monat
    if (b > S) then b:= S; fi:
    S:= S - b:   # jeweils noch verbleibende Schulden
    Z_ges:= Z_ges + Z:
    b_ges:= b_ges + b:
    P[t+1]:= [t, Z_ges]:
    printf(" %3.0f %9.2f %7.2f %9.2f %7.2f %9.2f\n",
               t,   S,   b,   b_ges,   Z,   Z_ges);
  od:

t     S      b    b_ges      Z    Z_ges
010000.000.000.000.000.00
19921.3378.6778.6758.3358.33
29842.2179.13157.7957.87116.21
39762.6279.59237.3857.41173.62
49682.5780.05317.4356.95230.57
59602.0580.52397.9556.48287.05
69521.0680.99478.9456.01343.06
79439.6081.46560.4055.54398.60
89357.6781.94642.3355.06453.67
99275.2582.41724.7554.59508.25
109192.3682.89807.6454.11562.36
119108.9883.38891.0253.62615.98
129025.1283.86974.8853.14669.12
138940.7684.351059.2452.65721.76
148855.9284.851144.0852.15773.92
158770.5885.341229.4251.66825.58
168684.7485.841315.2651.16876.74
178598.4086.341401.6050.66927.40
188511.5686.841488.4450.16977.56
198424.2187.351575.7949.651027.21
208336.3587.861663.6549.141076.35
218247.9888.371752.0248.631124.98
228159.0988.891840.9148.111173.09
238069.6989.411930.3147.591220.69
247979.7689.932020.2447.071267.76
257889.3190.452110.6946.551314.31
267798.3390.982201.6746.021360.33
277706.8291.512293.1845.491405.82
287614.7892.042385.2244.961450.78
297522.1992.582477.8144.421495.19
307429.0793.122570.9343.881539.07
317335.4193.662664.5943.341582.41
327241.2094.212758.8042.791625.20
337146.4494.762853.5642.241667.44
347051.1395.312948.8741.691709.13
356955.2695.873044.7441.131750.26
366858.8396.433141.1740.571790.83
376761.8496.993238.1640.011830.84
386664.2997.563335.7139.441870.29
396566.1698.123433.8438.881909.16
406467.4698.703532.5438.301947.46
416368.1999.273631.8137.731985.19
426268.3499.853731.6637.152022.34
436167.90100.433832.1036.572058.90
446066.88101.023933.1235.982094.88
455965.27101.614034.7335.392130.27
465863.07102.204136.9334.802165.07
475760.27102.804239.7334.202199.27
485656.87103.404343.1333.602232.87
495552.87104.004447.1333.002265.87
505448.26104.614551.7432.392298.26
515343.05105.224656.9531.782330.05
525237.21105.834762.7931.172361.21
535130.76106.454869.2430.552391.76
545023.69107.074976.3129.932421.69
554916.00107.705084.0029.302451.00
564807.67108.325192.3328.682479.67
574698.72108.965301.2828.042507.72
584589.13109.595410.8727.412535.13
594478.90110.235521.1026.772561.90
604368.03110.875631.9726.132588.03
614256.51111.525743.4925.482613.51
624144.33112.175855.6724.832638.33
634031.51112.825968.4924.182662.51
643918.03113.486081.9723.522686.03
653803.88114.146196.1222.862708.88
663689.07114.816310.9322.192731.07
673573.59115.486426.4121.522752.59
683457.44116.156542.5620.852773.44
693340.61116.836659.3920.172793.61
703223.09117.516776.9119.492813.09
713104.89118.206895.1118.802831.89
722986.01118.897013.9918.112850.01
732866.42119.587133.5817.422867.42
742746.14120.287253.8616.722884.14
752625.16120.987374.8416.022900.16
762503.48121.697496.5215.312915.48
772381.08122.407618.9214.602930.08
782257.97123.117742.0313.892943.97
792134.14123.837865.8613.172957.14
802009.59124.557990.4112.452969.59
811884.31125.288115.6911.722981.31
821758.31126.018241.6910.992992.31
831631.56126.748368.4410.263002.56
841504.08127.488495.929.523012.08
851375.85128.238624.158.773020.85
861246.88128.978753.128.033028.88
871117.15129.738882.857.273036.15
88986.67130.489013.336.523042.67
89855.43131.249144.575.763048.43
90723.42132.019276.584.993053.42
91590.64132.789409.364.223057.64
92457.08133.559542.923.453061.08
93322.75134.339677.252.673063.75
94187.63135.129812.371.883065.63
9551.72135.919948.281.093066.72
960.0051.7210000.000.303067.03

Die zur Zeit t bisher insgesamt gezahlten Zinsbeträge (Z_ges):

> t:= 't': Z_ges:= 'Z_ges':
  plot({seq(P[i], i = 1..97)},
    x = 0..96,
    style = point,
    symbol = CROSS,
    color = red,
    labels = [t, Z_ges]);

Z_ges in Abhängigkeit von t


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