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Anexo 4
RESPUESTAS
GRADIENTES ARITMÉTICOS PROBLEMA 1.-
VALOR FUTURO
Los pagos forman una sucesión aritmética, en donde la cantidad base es $1,300.00 y el gradiente es igual a $200.00. Datos: RP=$1,300.00 Ga=$200.00 n=12 i=30% anual =30/12=2.5% mensual
( ⁄)[( ⁄ )
⁄
] ⁄
Sustitución de Valores en la Fórmula:
(
) *( )
+
( ) *( )
+
( ) [
]
( ) [
]
( )[ ]
VALOR ACTUAL Datos: RP=$1,300.00 Ga=$200.00 n=12 i=30% anual =30/12=2.5% mensual
542
[( ⁄)[( ⁄ )
⁄
] ⁄] ( ⁄ )
*(
) *( )
+
+ ( )
*( ) *( )
+
+ ( )
[( ) [
] ] ( )
[( ) [
] ] ( )
[( )[ ] ]( )
[ ]( )
( )( )
PROBLEMA 2.-
VALOR FUTURO Datos: n = 30 mensualidades Mga=?
i= 35% cap. mensual Rp=$4,200.00 ga = $1,500.00
1
(1 / ) 1 *( )
/ / /
nga i m n gaMga Rp
i m i m i m
30$1,500.00 (1 .35/12) 1 30 *$1,500.00($4,200.00 )
.35/12 .35/12 .35/12Mga
30$1,500.00 (1 .029166666) 1 $45,000.00($4,200.00 )
.029166666 .029166666 .029166666Mga
30(1.029166666) 1($4,200.00 $51,428.5726) $1,542,857.178
.029166666Mga
1.369034242($4,200.00 $51,428.5726) $1,542,857.178
.029166666Mga
$55,628.5726 46.93831794 $1,542,857.178Mga
$2,611,111.627 $1,542,857.178Mga
$1,068,254.449Mga
543
VALOR ACTUAL
Datos: n = 30 mensualidades Mga= $1,068,254.449
i= 35% cap. mensual Rp=$4,200.00 ga = $1,500.00
1
(1 / ) 1 *( ) (1 / )
/ / /
nnga i m n ga
VA Rp i mi m i m i m
(1 / ) nVA Mga i m
30301500 (1 .35/12) 1 30*1500
(4,200 ) (1 .35/12).35/12 .35/12 .35/12
VA
301.369034242 45000(4,200 51,428.5726) (1.029166666)
.029166666 .029166666VA
30(55,628.5726) 46.93831794 1,542,857.178 (1.029166666)VA
2,611,111.627 1,542,857.178 (.422112936)VA
1,068,254.449 (.422112936)VA
$450,924.02222VA
PROBLEMA 3.-
VALOR FUTURO Datos: RP1: $35,000.00 Ga: $600.00 n: 10 i/m: 20% capitalizable: ( .20/12)= .016666
1
(1 / ) 1 *( )
/ / /
nga i m n gaMga Rp
i m i m i m
544
10
10
$600.00 (1 (.20 /12)) 1 10 *$600.00($35,000.00 )
.20 /12 .20 /12 .20 /12
(1 0.0166666) 1 10 *$600.00($35,000.00 $36,001.44)
0.0166666 0.0166666
(1.17973798) 1($71,001.44)
0.0166666
Mga
Mga
Mga
$6,000.00
0.0166666
($71,001.44) 10.78432199 $360,001.44
$765,702.39 $360,001.44
$405,700.95
Mga
Mga
Mga
VALOR ACTUAL Datos: RP: $35,000.00 Ga: $600.00 n: 10 i/m: 20% capitalizable: .20/12: .016666
⌈
⌉ ⌈(
)
⌉ ⌈
⌉ (
)
1010
1010
$600.00 (1 (.20 /12)) 1 10 *$600.00 .20$35,000.00 *(1 )12.20 /12 .20 /12 .20 /12
(1 0.0166666) 1 $6,000.00$35,000.00 $36,001.44 *(1.166666)
0.0166666 0.0166666
$71,001.
VAga
VAga
VAga
(1.17973798) 1 $6,000.0044 *(0.21405844)
0.0166666 0.0166666
$71,001.44 10.78432199 $360,001.44 *(0.21405844)
$765,702.39 $360,001.44 *(0.21405844)
$405,700.95*0.21405844
$86,843.71
VAga
VAga
Mga
Mga
545
GRADIENTES GEOMÉTRICOS PROBLEMA 1.-
Cuotas Anticipadas (Prepagables) con Gg:
Datos: n = 9 Mgg=?
i= 10% anual =
% semestral= 5% semestral = 0.00833333 mensual
Rp=$24,870.00 Gg = 3.5% semestral
(1 / )Si i m Gg
1
(1 / ) (1 )(1 / )
( / )
n n
g
i m GgMg Rp i m
i m Gg
9 9(1 0.05/ 6) (1 0.035)
$24,870(1 0.05/ 6)(0.05/ 6) 0.035
gMg
9 9(1.00833333) (1.035)
$24,870.00(1.008333333)(0.008333) .035
gMg
1.077549192 1.362897353$25,077.24999
.026667gMg
0.285348161$25,077.24999
.026667gMg
$25,077.24999 10.70042228gMg
$268,337.1646gMg
546
TABLA DE DESPEJES Valor Actual del Rp
Valor de “n” plazo
Fórmula original:
(1 / )Si i m Gg
1
(1 / ) (1 )(1 / )
( / )
n n
g
i m GgMg Rp i m
i m Gg
Despeje:
1(1 / ) (1 )
(1 / )( / )
g
n n
MgRp
i m Ggi m
i m Gg
Datos: n = 9 Mgg= 268,337.1646
i= 10% anual =
% semestral= 5% semestral =
0.00833333 mensual Rp=? Gg = 3.5% semestral
19 9
$268,337.1646
(1 0.05/ 6) (1 .035)(1 .05/ 6)
(.05/ 6) .035
Rp
1
$268,337.1646
(1.077546018) (1.362897353)(1.0083333)
(.008333) .035
Rp
1
$268,337.1646
( 0.285351335)(1.0083333)
( 0.026667)
Rp
1
$268,337.1646
(1.0083333) 10.70054131Rp
1
$268,337.1646
(10.78971213)Rp
1$24,869.72417 Rp
Fórmula original:
1
1 1 / *( / ) 01 /
x x MggGg i m i m Gg
Rp i m
Se tiene que satisfacer la fórmula:
$268,337.16461 .035 1 .05 / 6 *(.05 / 6 .035) 0
$24,870.00 1 .05 / 6
x x
A prueba y error utilizamos para “x”= 8, 10 respectivamente y obtenemos:
8 8 $268,337.16461 .035 1 .05 / 6 *(.05 / 6 .035) 0
$24,870.00 1 .05 / 6
(1.316809037) 1.068643858 10.70042228*( .026666666) 0
(1.316809037) 1.068643858 0.285344594 .037179415
10 10 $268,337.16461 .035 1 .05 / 6 *(.05 / 6 .035) 0
$24,870.00 1 .05 / 6
(1.410598761) 1.086528801 10.70042228*( .026666666) 0
1.410598761 1.086528801 0.285344594 .038725366
“n” está entre 8 y 10
547
Cuotas Pospagables (vencidas) con Gg: Datos: n = 9 Mgg=?
i= 10% anual =
% semestral= 5% semestral = 0.00833333 mensual
Rp=$24,870.00 Gg = 3.5% semestral De la Fórmula:
(1 / )Si i m Gg
1
(1 / ) (1 )(1 / )
( / )
n n
g
i m GgMg Rp i m
i m Gg
Se Modifica:
(1 / )Si i m Gg
1
(1 / ) (1 )
( / )
n n
g
i m GgMg Rp
i m Gg
9 9(1 0.05/ 6) (1 0.035)$24,870.00
(0.05/ 6) 0.035gMg
9 9(1.0083333) (1.035)$24,870.00
(0.0083333) 0.035gMg
(1.07754903 1.362897353$24,870.00
.0266667gMg
0.28534323$24,870.00
.0266667gMg
$24,870.00 10.70054874gMg
$266,122.6471gMg
TABLA DE DESPEJES Valor Actual Rp1
Valor de “n” plazo
Fórmula original:
(1 / )Si i m Gg
1
(1 / ) (1 )
( / )
n n
g
i m GgMg Rp
i m Gg
Fórmula Original :
1
1 1 / *( / ) 0x x Mgg
Gg i m i m GgRp
548
Despeje:
1(1 / ) (1 )
( / )
g
n n
MgRp
i m Gg
i m Gg
Datos: n = 9 Mgg=$266,122.6471
i= 10% anual=
% semestral= 5% semestral =
0.00833333 mensual Rp=? Gg = 3.5%
19 9
266122.6471
(1 .05/ 6) (1 .035)
(.05/ 6) .035
Rp
1
266122.6471
(1.077549224) (1.362897353)
(.026666)
Rp
1
266122.6471
(.285348129)
(.026666)
Rp
1
266122.6471
10.70082236Rp
1
24869.36407 Rp
Se tiene que satisfacer la fórmula:
$266,122.6471
1 .035 1 .05 / 6 *(.05 / 6 .035) 0$24,870.00
x x
A prueba y error utilizamos para “x”= 8, 10 respectivamente y obtenemos:
8 8 $266,122.6471
1 .035 1 .05 / 6 *(.05 / 6 .035) 0$24,870.00
(1.316809037) 1.068643858 10.70054874*( .026666666) 0
(1.316809037) 1.068643858 0.285347966 .037182787
10 10 $266,122.6471
1 .035 1 .05 / 6 *(.05 / 6 .035) 0$24,870.00
(1.410508761) 1.086528801 10.70054874*( .026666666) 0
1.410508761 1.068643858 0.285347966 .056516937
“n” está entre 8 y 10
549
PROBLEMA 2.-
Cuotas Anticipadas (Prepagables) con Gg: Datos: n = 18 mensualidades Mgg=?
i= 27% nominal con capitalización mensual Rp=$2,700.00 Gg = 4.3%
(1 / )Si i m Gg
1
(1 / ) (1 )(1 / )
( / )
n n
g
i m GgMg Rp i m
i m Gg
18 18(1 .27 /12) (1 .043)$2,700.00(1 .27 /12)
(.27 /12) .043g
Mg
18 18(1.0225) (1.043)$2,700.00(1.0225)
(.0225) .043g
Mg
1.492587156 2.133622348$2,760.75
.0205g
Mg
.641035192$2,760.75
.0205g
Mg
$2,760.75 31.27000937gMg
$86,328.67836g
Mg
TABLA DE DESPEJES Valor Actual del Rp
Valor de “n” plazo
Fórmula original:
(1 / )Si i m Gg
1
(1 / ) (1 )(1 / )
( / )
n n
g
i m GgMg Rp i m
i m Gg
Despeje:
1(1 / ) (1 )
(1 / )( / )
g
n n
MgRp
i m Ggi m
i m Gg
Fórmula:
1
1 1 / *( / ) 01 /
x x MgaGg i m i m Gg
Rp i m
Se tiene que satisfacer la fórmula:
$86,328.678361 .043 1 .27 /12 *(.27 /12 .043) 0
$2,700.00 1 .27 /12
x x
A prueba y error utilizamos para “x”= 17, 19 respectivamente y obtenemos:
550
Datos: n = 18 mensualidades Mgg=$86,328.67836 i= 27% nominal con capitalización mensual Rp=? Gg = 4.3%
118 18
$86,328.67836
(1 .27 /12) (1 .043)(1 .27 /12)
(.27 /12) .043
Rp
118 18
$86,328.67836
(1.0225) (1.043)(1.0225)
(.0225) .043
Rp
1
$86,328.67836
1.492587156 2.133622348(1.0225)
.0205
Rp
1
$86,328.67836
.641035192(1.0225)
.0205
Rp
1
$86,328.67836
(1.0225) 31.27000937Rp
1
$86,328.67836
31.97358458Rp
1
$2,700.00 Rp
17 17 $86,328.678361 .043 1 .27 /12 *(.27 /12 .043) 0
$2,700.00 1 .27 /12
(2.045659011) 1.45974294 31.27000937*( .0205) 0
(2.045659011) 1.45974294 .641035192 .055119121
19 19 $86,328.678361 .043 1 .27 /12 *(.27 /12 .043) 0
$2,700.00 1 .27 /12
$86,328.67836
2.225368109 1.526170367 *( .0205) 02,760.75
2.225368109 1.526170367 31.27000764*( .0205) 0
2.225368109 1.526170367 .641035156 .058162586
“n” está entre 17 y 19
551
Cuotas Pospagables (vencidas) con Gg: Datos: n = 18 mensualidades Mgg=?
i= 27% nominal con capitalización mensual Rp=$2,700.00 Gg = 4.3%
De la Fórmula:
(1 / )Si i m Gg
1
(1 / ) (1 )(1 / )
( / )
n n
g
i m GgMg Rp i m
i m Gg
Se Modifica:
(1 / )Si i m Gg
1
(1 / ) (1 )
( / )
n n
g
i m GgMg Rp
i m Gg
18 18(1 .27 /12) (1 .043)$2,700.00
(.27 /12) .043g
Mg
18 18(1.0225) (1.043)$2,700.00
(.0225) .043g
Mg
(1.492587156 2.133622348$2,700.00
.0205g
Mg
.641035192$2,700.00
.0205g
Mg
$2,700.00 31.27000937gMg
$84,429.02529g
Mg
TABLA DE DESPEJES Valor Actual Rp1
Valor de “n” plazo
Fórmula original:
(1 / )Si i m Gg
1
(1 / ) (1 )
( / )
n n
g
i m GgMg Rp
i m Gg
Despeje:
Fórmula Original:
1
1 1 / *( / ) 0x x Mga
Gg i m i m GgRp
Se tiene que satisfacer la fórmula:
552
1(1 / ) (1 )
( / )
g
n n
MgRp
i m Gg
i m Gg
Datos: n = 18 mensualidades
Mgg= 84,429.02529
i= 27% cap. mensual Rp=? Gg = 4.3%
118 18
$84,429.02529
(1 .27 /12) (1 .043)
(.27 /12) .043
Rp
118 18
$84,429.02529
(1.0225) (1.043)
(.0225) .043
Rp
1
$84,429.02529
1.492587156 2.133622348
.0205
Rp
1
$84,429.02529
.641035192
.0205
Rp
1
$84,429.02529
31.27000937Rp
1
$2,700.00 Rp
$84, 429.02529
1 .043 1 .27 /12 *(.27 /12 .043) 0$2,700.00
x x
A prueba y error utilizamos para “x”= 17, 19 respectivamente y obtenemos:
17 17 $84, 429.02529
1 .043 1 .27 /12 *(.27 /12 .043) 0$2,700.00
(2.045659011) 1.45974294 31.27000948*( .0205) 0
(2.045659011) 1.45974294 .641035194 .055119123
19 19 $84,429.02529
1 .043 1 .27 /12 *(.27 /12 .043) 0$2,700.00
$84,429.02529
2.225368109 1.526170367 *( .0205) 02,700.00
2.225368109 1.526170367 31.27000948*( .0205) 0
2.225368109 1.526170367 .641035194 .062629245
“n” está entre 17 y 19
553
GRADIENTES ARITMETICO-GEOMETRICO PROBLEMA 1.-
[(
)
⌈ (
)
⌉] [ ⌈(
) ( )
(
) ⌉ ]
Datos: A1: 1.5 Gg: .17 n: 8 i: 15% Capitalización mensual, por lo que sería .15/12= 0.0125
[( ) ⌈ ( )
⌉] [
⌈( ) (
)
( ) ⌉ ]
[ ⌈
⌉] [ ⌈
( )
⌉ ]
[ ⌈ ( )⌉] [ ⌈
⌉ ]
[ ⌈ ⌉] [ ⌈ ⌉]
[ ]
PROBLEMA 2.-
Datos: A1:$5’500,000.00 =5.5 Gg: $850,000.00 =.85 n: 40 i: 19.65% nominal con capitalización mensual, por lo que sería .1965/12= 0.016375
[(
)
⌈ (
)
⌉] [ ⌈(
)
( )
(
) ⌉]
554
[( ) ⌈ ( )
⌉] [
⌈( ) ( )
( ) ⌉]
[( ) ⌈
⌉] [ ⌈
( )
⌉]
[( ) ⌈ ( )⌉] [ ⌈ ( )
⌉]
[( ) ⌈ ( )⌉] [ ⌈
⌉]
[( ) ⌈ ( )⌉] [ ⌈ ⌉]
[ ⌈ ⌉] [ ]
