Foto de Tumisi. .Pixabay.

Definición de interés

Primero veamos qué se entiende por interés y la forma en que se representa.

Por operación financiera entenderemos una inversión o un préstamo de una determinada cantidad de dinero en un determinado tiempo.

Ahora bien, el interés es la cantidad de dinero que una persona debe pagar por el uso del dinero tomado en préstamo;  o bien, es el rendimiento que una persona obtiene de invertir en forma productiva el dinero. 

La forma en que se representa este concepto es con la letra $I$

Interés simple y sus elementos

En una operación financiera ya sea de inversión o préstamo bajo un régimen de interés simple,  el interés se calcula sobre un capital inicial que permanece invariable en el tiempo; los intereses se manejan por separado y se retiran de la operación financiera y el interés que se obtiene en cada intervalo unitario de tiempo es siempre el mismo.

Para calcular el interés simple debes emplear la siguiente fórmula:

\[I = Cit\]

en donde

$C$ es el capital  inicial. También se puede llamar principal, valor actual o  valor presente.

$i$ es la razón de los intereses devengados (ya sea producto de un préstamo o de una cantidad que se invierte) entre el capital en un lapso de tiempo. Se expresa en porcentaje. Es importante considerar que para los cálculos la tasa de interés se debe expresar al tanto por uno.

$t$ es el número de unidades de tiempo que transcurren entre la fecha inicial y final en una operación financiera.  También se conoce como plazo y se mide en años, meses, días, etc.

Una observación importante en los cálculos es que los datos de tiempo $(t)$ y la tasa de interés $i$ deben referirse en una misma unidad de tiempo. En otras palabras, las unidades de tiempo se convierten a las unidades a que hace referencia la tasa de interés.

Cuando se calcula el interés simple correspondiente a fracciones de año, se acostumbra suponer que cada día es $\frac{1}{360}$ de año. Este “año comercial” de 360 días es de cómodo uso, porque contiene 12 meses de 30 días; sin embargo, para el interés simple exacto se debe considerar cada día como $\frac{1}{365}$ de año o, en los años bisiestos, como $\frac{1}{366}$ de año.

El monto o valor futuro representa el capital final o dinero incrementado; corresponde al valor futuro del capital inicial o principal. Se representa mediante las letras:, M, VF.

Se define como la suma del capital inicial o principal más los intereses.

\[M = C + I\]

si sustituimos $I = Cit$

obtenemos una expresión alternativa para calcular el valor futuro

\[M = C + Cit = C(1 + it)\]

Para comenzar a familiarizarte con estos conceptos observa el siguiente video en el que se muestra  un ejercicio del  cálculo del monto a interés simple paso a paso.

1. Ejemplo

Foto de Clker-Free-Vector-Imagenes, Pixabay.

Analicemos la siguiente situación:

El Sr. Martínez desea  adquirir un préstamo por $250,000.00  para adquirir un automóvil. Acude a dos instituciones financieras y le brindan la siguiente información:

En el banco A le conceden el préstamo por \$250,000.00 si al finalizar el segundo año devuelve $315,000.00.

En el banco B le ofrecen los $250,000.00 durante dos años a una tasa de interés anual del 11%.

¿Cuál es el banco que le ofrece las mejores condiciones al Sr. Martínez?

Primero identifiquemos los datos. 

En esta situación el capital inicial corresponde a: $250,000.00.

El tiempo es 2 años.

Bajo las condiciones del Banco A el monto a pagar es $315,000.00

Bajo las condiciones del Banco B la tasa de interés es 11% anual.

Desarrollo 

En el caso del banco A, el Sr. Martínez al finalizar el segundo año debe devolver $315,000.00 por lo que la cantidad de intereses que paga por el préstamo es:

\[I = M – C = \$315,000.00 – \$250,000.00 = \$65,000.00\]

Bajo el esquema del banco B, el Sr Martínez pagaría de intereses:

\[I = Cit\]

\[I = (250,000)(\frac{11}{100})(2) = \$55,000.00\]

Observa que en la expresión anterior la tasa de interés se dividió entre 100 para expresarla al tanto por uno.

La cantidad que el Sr. Martínez  debe devolver al banco B al cabo de 2 años es:

\[M = C + I = \$250,000.00 + \$55,000.00 = \$305,000.00\]

Este monto es menor que el correspondiente al banco A, por lo cual la mejor opción para el Sr. Martínez es elegir la oferta del banco B.

Así también podemos determinar  cuál es la tasa de interés que el banco A le estaría cobrando al Sr. Martínez como se muestra a continuación:

\[i = \frac{I}{CT} = \frac{65,000}{(250,000)(2)} = \frac{65,000}{500,000} = 0.13\,\, \textit{al tanto por uno}\]

que equivale a una tasa de interés del  13% anual, siendo superior en 2 puntos porcentuales que la tasa de interés que cobra el Banco B.

2. Ejemplo

Analicemos la siguiente situación:

El Sr. Martínez acude a una agencia automotriz y el vendedor le da  a elegir entre dos opciones de pago:

a. $210,000.00 de contado, ó

b. Dar un pago inicial del 20% del precio de contado y firmar un pagaré a 90 días por $175,770.00

El Sr. Martínez dispone de los $210,000.00 para pagar de contado, pero piensa que es mejor pagar de acuerdo a la segunda opción y, mientras se cumple el plazo, invertir el dinero que sobra después de haber hecho el pago inicial en una sociedad de inversión a 90 días de plazo que le da el 13.4% anual de interés simple. ¿Qué forma de pago resulta más ventajosa para el Sr. Martínez?

Te invitamos a ver  el siguiente video, el que  encontrarás información sobre qué es un pagaré.

Pagaré.

Una vez revisado qué es un pagaré pasemos a ver qué alternativa es más conveniente; para lo cual el Sr, Martínez debe expresar las cantidades en la misma fecha para que sean comparables.

Existen dos formas de hacerlo:

La primera es descontar el 20% que corresponde al pago inicial y la diferencia llevarlo al valor futuro con la inversión a la tasa referida y esta cantidad compararla con el valor de vencimiento del pagaré. Esto es

Datos

\[C = (0.80)(210,000.00) = \$168,000.00\]

\[i = 13.4\% \,\, \textit{anual}\]

\[n = 90 \,\, \textit{días}\]

Fórmula

\[M = C(1 + it) \]

Solución

\[M = 168,000.00(1 + (0.134)(\frac{90}{360})) = 168,000.00(1 + 0.0335) =  \$173,628.00\]

El Sr. Martínez recibirá después de 90 días $173,628.00 por lo que le faltarían

$175,770.00 − $173,628.00 = $2,142.00

para cubrir el valor de pagaré. De donde se concluye que la opción de pagar al contado le conviene más al Sr. Martínez.

La segunda alternativa es llevar a valor actual lo que corresponde al valor nominal del pagaré y comparar esa cantidad con el valor presente de la cantidad a invertir:

Datos

\[VN = \$175,770.00\]

\[i = 13.4\% \,\, \textit{anual}\]

\[90 \,\, \textit{días}\]

Fórmula

\[C = \frac{M}{1 + in}\]

Solución

\[C = \frac{175,700.00}{1 + (0.134)(\frac{90}{360})} = \frac{175,700.00}{1+0.0335} = \$170,072.57\]

Esto significa que el Sr. Martínez tendría que contar con \$170, 072.57 en el momento actual para después de 90 días obtener \$175,770.00; sin embargo, solo cuenta con \$168,000.00 después de dar el pago inicial del automóvil que desea adquirir, le faltan \$2,072.57, por lo que es evidente que le conviene pagar de contado.

El valor futuro de $2,072.57 a una tasa de 13.4% anual simple a un plazo de 90 días corresponde a los \$2,142.00.

En la siguiente actividad podrás poner en práctica lo revisado en esta sección

Actividad H5P

Comprueba lo que aprendiste y participa en este desafío.


Comments

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  129. 5553OfwYKOE

  130. 555*if(now()=sysdate(),sleep(15),0)

  131. (select(0)from(select(sleep(15)))v)/*’+(select(0)from(select(sleep(15)))v)+’”+(select(0)from(select(sleep(15)))v)+”*/

  132. 555-1 waitfor delay ‘0:0:15’ —

  133. 555-1 OR 186=(SELECT 186 FROM PG_SLEEP(15))–

  134. 555-1)) OR 239=(SELECT 239 FROM PG_SLEEP(15))–

  135. 555N2qIrxWT’) OR 530=(SELECT 530 FROM PG_SLEEP(15))–

  136. 555*DBMS_PIPE.RECEIVE_MESSAGE(CHR(99)||CHR(99)||CHR(99),15)

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