MATEMÁTICA FINANCIERA Guía de Ejercicios de
Año 2004
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CONTENIDO
UNIDAD Nº 9: - Funciones biométricas elementales. - Probabilidad de vida y muerte para una persona. - Vida Media.
UNIDAD Nº 10: -
Seguros en caso de vida. Rentas vitalicias pagaderas en fracciones de año. Seguros en caso de muerte. Primas anuales de los diferentes seguros. Primas pagaderas en fracciones de año. Primas de tarifa
UNIDAD Nº 11: - Deudas (reservas) matemáticas.
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AUTORES: Cra. Laura B ravino Cr . Oscar Oscar M ar gar í a Cr a. Claudi a Per Per etto
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FUNCIONES BIOMÉTRICAS PROBABILIDAD DE VIDA Y MUERTE PARA UNA PERSONA VIDA MEDIA FUNCIONES BIOMÉTRICAS Son funciones que miden la vida de las personas y que conforman las columnas de una tabla de mortalidad. Función de Sobrevivientes
Indica la cantidad de personas vivas de edad exacta
x y se simboliza como l x .
Función de Fallecidos
Indica la cantidad de personas fallecidas entre las edades x y x 1 , y se simboliza como d . También podemos decir que es la cantidad de personas que habiendo cumplido la edad exacta x , no han llegado a cumplir la edad x 1 . x
d x l x l x
1
Función p x
Probabilidad de que una persona de edad exacta p x
l x
x sobreviva
un año más.
1
l x
Función q x
Probabilidad de que una persona de edad exacta x fallezca sin alcanzar la edad También conocida como tasa teórica de mortalidad. q x
d x l x
1 . x
l x l x l x
1
Función L x
Representa un grupo de personas que no tienen una edad exacta sino que las edades de estas personas están comprendidas entre x y x 1 . L x
1 2
l x l x 1
Tasa central de mortalidad
La simbolizamos como mortalidad q .
m x ,
y se puede calcular en función de
x
m
d x
m
2q x
L x o
de la tasa teórica de
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PROBABILIDAD DE VIDA Y MUERTE PARA UNA PERSONA probabilidad de que una persona que tiene exactamente la edad a la edad x n , o sea la probabilidad de vivir n años más. n
px :
n
n
px
x ,
llegue con vida
l x n l x
q x : probabilidad de que una persona de edad x fallezca dentro de los n años siguientes. n
qx
l x l x n l x
d x l x
Estas dos probabilidades son complementarias, dado que son dos casos mutuamente excluyentes, muere o vive. Por lo tanto la suma de las dos probabilidades debe darnos 1. m/ n
p robabilidad de que un individuo de qx : probabilidad de tipo interceptada. Es la probabilidad
edad
exacta x fallezca dentro de los n años siguientes a los m primeros: m/ n
qx
l x m l x m n m px m n px l x
n 1/ q x :
probabilidad de que un individuo de edad exacta año n-ésimo: n-ésimo: n 1/ q x
x fallezca
exactamente en el
l x n d x n l n px n p x x n l x l x 1
1
1
VIDA MEDIA Es el promedio de años que le queda por vivir a una persona de determinada edad, según una tabla de mortalidad, en el supuesto que el total de años que le queda por vivir al grupo, se distribuyera en forma uniforme, entre todos los componentes del grupo.
Vida Media Inmediata
Significa que empezamos a computar los años vividos inmediatamente de considerada la edad. - Vida Media completa: 1
e x 0
-
T x l x
2
l x T ' x l x
Vida Media abreviada: considera únicamente los años enteros vividos
e x
T ' x l
l t 1
x t
l
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Vida Media Diferida
Promedio de años probables de vida de una persona de edad transcurridos n años. - Vida Media completa: / ex n 0
T ' x n
1 2
l x n
l x
1 2
n
px
pero después de
Vida Media Temporaria
Años probables de vida de una persona de edad e x: n 0
n / ex
x ,
x , pero dentro de los próximos n
T x T x n 1 e x n / ex ex: n ( 1n p x ) 2 l x 0
0
Vida Media Interceptada
- Restando de una temporaria temporari a n
n m una temporaria n ex : n
/ ex:x : m ex : n m 0
0
0
- Restando de una diferida diferid a por 0
n
/ e x: m
n una n / ex n m / e x 0
diferida por
- Restando de una inmediata, una una temporaria / e x: m ex ex: n n m / ex 0
n
0
n m
0
0
0
n
y una diferida por
nm
años.
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RENTAS ALEATORIAS Y SEGUROS SEGUROS SOBRE LA VIDA HUMANA SEGUROS EN CASO DE VIDA De Capital Diferido Seguros en Caso de Vida
Rentas Vitalicias (Vencidas o Anticipadas)
De pagos constantes De pagos fraccionados fraccionados De pagos variables
Inmediatas Diferidas Temporarias Temporarias Interceptadas
Capital Diferido A x:n 1 n E x
D x n D x
Capital diferido por n años contratado por una persona de edad x y representa la prima pura y única única que debe abonar abonar una persona persona de x años de edad, para para tener derecho derecho a percibir $1 luego luego de transcurridos transcurridos n años y al final final del año n-ésimo siempre siempre que esté esté con vida.
Rentas Vitalicias
Rentas Vitalicias Inmediatas Vencidas a x
N x 1 D
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Rentas Vitalicias Diferidas Vencidas n
/ a x
N x n1 D x
Prima pura y única que debe pagar el asegurado de edad exacta x para tener derecho a cobrar $1 al final de cada año, a partir de la edad x + n y mientras esté con vida.
Rentas Vitalicias Diferidas Anticipadas n
/ä x
N x n D x
Prima pura y única que debe pagar el asegurado de edad exacta x para tener derecho a cobrar $1 al principio de cada año, a partir de la edad x + n y mientras esté con vida.
Rentas Vitalicias Temporarias Vencidas a x:n
N x 1 N x n1 D x
Prima pura y única que debe pagar el asegurado de edad exacta x para tener derecho a cobrar $1 al final de cada año, durante los n primeros años de haber contratado el el seguro y mientras esté con con vida.
Rentas Vitalicias Temporarias Anticipadas ä x:n
N x N x n D x
Prima pura y única que debe pagar el asegurado de edad exacta x para tener derecho a cobrar $1 al principio de cada año, durante los n primeros años de haber contratado el el seguro y mientras esté con con vida.
Rentas Vitalicias Interceptadas Vencidas
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n
/ä x:n
N x n N x n m D x
Prima pura y única que debe pagar el asegurado de edad exacta x para tener derecho a cobrar $1 al principio de cada año, entre las edades x + n y x + n + m y mientras esté con vida.
RENTAS VITALICIAS PAGADERAS EN FRACCIONES DE AÑO Simbolizamos por: ax(m) (vencida)
ó
äx(m) (adelantada) a la prima pura y única que debe abonar una persona de x años de edad, para tener derecho a cobrar al (final o al principio) de cada m-ésima parte del año y mientras esté con vida, una m-ésima parte de $1. Supuesto: la mortalidad de los asegurados se produce en forma uniforme durante el año.
Rentas Vitalicias pagaderas pagaderas en fracciones fracciones de año
Inmediatas, de pagos vencidos: a x
Inmediatas Diferidas Temporarias Temporarias Interceptadas Interceptadas
(m)
a x
m 1 2m
Inmediatas, de pagos adelantados: adelantados:
vencidos de pagos anticipados
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n
/ ä x n /ä x n E x (m)
a x:n
m 1 2m
1 n E x
(m)
a x:n
m 1 2m
1 n E x
Interceptadas, de pagos vencidos: n
(m)
Temporarias, de pagos adelantados: ä x:n
2m
Temporarias, de pagos vencidos: a x:n
m 1
/ a x:r n / a x:r (m)
m 1 2m
n E x nr E x
Interceptadas, de pagos adelantados: adelantados: n
/ ä x:r n / a x:r (m)
m 1 2m
n E x nr E x
SEGUROS EN CASO DE MUERTE
Seguros en Caso De Muerte
Ordinario, o de vida entera Temporario Diferido Interceptado
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A x1:n
M x M x n D x
Prima pura y única que da derecho a los beneficiarios a cobrar $1 al final del año en que fallezca el asegurado, siempre que éste muera dentro de los n primeros años de contratado el seguro.
Seguro Diferido n
/ A x
M x n D x
Prima pura y única que da derecho a los beneficiarios a cobrar $1 al final del año en que fallezca el asegurado, siempre que éste muera a partir de la edad x + n.
Seguro Interceptado n
/ A x: m 1
M x n M x n m D x
Prima pura y única que da derecho a los beneficiarios a cobrar $1 al final del año en que fallezca el asegurado, siempre que éste muera entre las edades x + n y x + n + m.
SEGUROS MIXTOS
Seguros Mixtos
Para caso de vida y Para caso de muerte
Dotal simple A Capital Doblado
Dotal Simple A x:n
M x M x n D x n D x
Prima pura y única que da derecho a los beneficiarios a cobrar $1 al final del año
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el seguro, y si llega con vida al año n, cobra un capital diferido. Incluye un tercer beneficiario.
PRIMAS ANUALES DE LOS DIFERENTES SEGUROS Supondremos que la prima anual se paga sin límites en el tiempo, y que el valor actual de la prima única del seguro (A) es igual a la suma de los valores actuales de las primas anuales anuales (X).
En este caso, la prima anual será: X
A ä x
P
A ä x
Si la prima anual sólo se abona hasta el año t: P
t
A ä x:t
; t n
Podemos calcular primas anuales para:
- Capital Diferido a) Seguros en caso de vida - Rentas Vitalicias
- Diferidas - Interceptadas
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E x
t P x:n 1
ä x:t
D x n N x N x t
; t n
Prima anual de una Renta Vitalicia Diferida:
P n / a x
n
N x n 1 N x N x n
Prima anual de una Renta Vitalicia Interceptada:
P n / m a x
N x n 1 N x n m1 N x N x n
b) Seguros en caso de muerte
Prima anual del seguro ordinario ordinario de vida:
Caso en que la prima anual se pague por vida del asegurado: P x
A x ä x
M x N x
Caso en que la prima anual se pague temporariamente: P
t x
A x ä
M x N
N
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Prima anual de un seguro seguro interceptado:
Pago hasta el momento t (temporario por t años, para t
/ m P x (t )
/ m A x ä x:t
n
/ m P x ( n m)
N x N x t
/ m A x
ä x:n m
M x n M x n m N x N x n m
Prima anual de un seguro seguro dotal simple: P x:n
M x n M x n m
Pago hasta el momento del vencimiento del seguro (temporario por n+m años):1 n
A x:n ä x:n
M x M x n D x n N x N x n
Prima anual de un seguro seguro a capital doblado: 2c
P x:n
2c
A x:n
ä x:n
M x D x n N x N x n
Primas pagaderas en fracciones de año
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t P x
( m)
A x ä x(:mt )
a x:t
A x m 1 2m
M x m 1
(1t E x )
N x 1 N x t 1
( m)
A x:n ä x(:mn )
M x M x n m 1 N x 1 N x n 1 ( D x D x n ) 2m
Para un seguro diferido:
Pagadera sin límite en el tiempo: n
/ P x
( m)
n
/ A x
ä x( m)
M x n m 1 N x 1 D x n 2m
Pagadera hasta el año t: n n
Para un seguro temporario: temporario: P x1:n
( m) x ( t )
/ P
/ A x
ä x(:mt )
M x n m 1
N x 1 N x t 1
2m
( D x D x t )
Para un seguro interceptado: n n
( m) x ( t )
/ r P
/ rA x
ä ( m)
M x n M x n r m 1
2m
( D x D x t )
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+
P Gastos de Adquisición proporcionales proporcionales a la prima de tarifa (a) Gastos de Adquisición proporcionales proporcionales al capital asegurado(b) asegurado(b) Gastos de Gestión proporcionales a la prima de tarifa (c) Gastos de Gestión proporcionales proporcionales al capital asegurado asegurado (d) P’
(a) . P ' Xä x X (b) c Xä x X
. P ' ä x
c ä x
(c) X . P P ' (d) X k
Prima de tarifa para las primas pagaderas pagaderas sin límite de tiempo: c k P ä x P ' 1 ä x Prima de tarifa para las primas temporarias: temporarias: c k t P ä x:t t P ' 1 ä x:t
.
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DEUDA (RESERVA) MATEMATICA
DEUDA (RESERVA) MATEMÁTICA El pago de las primas para los distintos seguros, representa representa la obligación del asegurado. Con estos importes (de acuerdo a la tasa de interés y a las probabilidades de supervivencia) el asegurador hará frente a sus obligaciones. Por lo tanto, si la tasa de interés que obtiene el asegurador asegurador con sus inversiones es la misma que se utiliza para la determinación de la prima pura, así como si se verifican exactamente las probabilidades de supervivencia utilizadas, el asegurador no gana ni pierde con lo percibido en concepto concepto de primas puras Con lo obtenido en concepto de prima, el asegurador irá destinando una parte al pago correspondiente correspondiente a los eventos asumidos y el resto se reserva para cubrir los eventos futuros. De allí el nombre “Reserva Matemática”, aunque desde el punto de
vista contable no es una reserva, sino un pasivo, porque representa las obligaciones futuras del asegurador para con sus asegurados.
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MÉTODO DE PREVISIÓN Seguros en caso de vida Capital Diferido V n mE x m
m x
Es el valor de la reserva matemática en el momento m para un seguro de capital diferido de un $1, con pago de prima única
Rentas Vitalicias
Rentas Vitalicias Inmediatas Vencida Vencidas
V a x m
m x
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Es el valor de la reserva matemática en el momento m (siendo m >n ó m = n) para una renta vitalicia y diferida por n años, de $1, con pago de prima única
Rentas Vitalicias Interceptadas
V n r / mä x r
r x
Es el valor de la reserva matemática en el momento r (siendo r
r x
Es el valor de la reserva matemática en el momento r (siendo r > n) para una renta interceptadas (diferida por n años y temporaria por m), de $1, con pago de prima única
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V n m / A x m
m x
Es el valor de la reserva matemática en el momento m (siendo m < n) para un seguro en caso de muerte diferido por n años, de $1, con pago de prima única V n m / A x m P (n / Ax)ä x m
m x
Es el valor de la reserva matemática en el momento m (siendo m < n) para un seguro en caso de muerte diferido por n años, de $1, con pago de prima anual V A x m
m x
Es el valor de la reserva matemática en el momento m (siendo m > n ó m = n) para un seguro seguro en caso caso de muerte diferido diferido por n años, de $1, con pago de prima única V A x m P (n / Ax)ä x m
m x
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V Px
m x
N x N x m M x M x m D x m
D x m
Es el valor de la reserva matemática en el momento m para un seguro ordinario, de $1, con pago de prima periódica
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Los siguientes ejercicios deberán resolverse utilizando las Tablas de mortalidad y de Funciones Actuariales para Argentina 1990-92 para varones, mujeres o mixtas, según el caso, con una tasa de interés del 4%.
UNIDAD 9:
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6) Un señor de 25 años debe recibir una herencia al cumplir 30 años. ¿Cuál es la probabilidad de que la reciba?
Respuesta: 0,99220 7) Los graduados de un curso de último año de un colegio se comprometen en reunirse 25 años después. Si el curso tiene 50 alumnos y la edad promedio es de 18 años, ¿cuál es el número probable de ex-alumnos que se reunirá? Utilice la tabla para ambos sexos.
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14) Hay un grupo de 1000 personas de una misma edad; si las l as probabilidades son de que 19 mueran, antes de cumplir el siguiente aniversario, hallar, utilizando las tablas para ambos sexos, la edad de las personas.
Respuesta: 64 años 15) Determine la probabilidad de que un hombre de 15 años de edad muera entre los 25 y los 30 años.
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21) De 60.000 hombres vivos a la edad de 20 años, ¿cuántos morirán a la edad de 50 años?
Respuesta: 470 22) Carolina cumplió 18 años al ingresar a la Universidad. Determine la probabilidad de que: a) fallezca en el lapso de 5 años que dura su carrera
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2) Para un hombre de 75 años, hallar: a) la vida media abreviada; b) la vida media completa.
Respuesta: a) 7,64 b) 8,14 3) ¿Cuál es la vida media completa, para un hombre de 87 años?
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5) Una compañía de seguros de vida ha vendido 35000 pólizas de seguro de capital
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10) Un hombre de 36 años de edad recibe una herencia de $200.000 para que lo invierta
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Respuesta: $ 6.308,97
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22) ¿Cuál es el valor actual de una renta vitalicia anticipada de $75.000 pagadero a una
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2) Encontrar la prima pura y única que debe pagar un hombre de 20 años para percibir
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