[11]
Question
3
Consider
a
30
year
term
insurance
with
death
benefit
of
$100,000
payable
immediately
on
death
sold
to
a
life
aged
25
exact.
Premiums
are
payable
monthly
in
advance
throughout
life
but
for
a
maximum
of
240
payments.
Expenses
are
20%
of
the
premiums
in
the
first
year
and
1%
of
all
subsequent
premiums.
Mortality
follows
AM92
Select.
The
interest
rate
is
4%
per
annum
effective.
[8]
(a)
Calculate
the
monthly
premium.
3
b}
Find
the
probability
that
the
death
benefit
1s
paid
AND
that
all
240
premiums
are
paid.

¥
I
F
P
State
any
assumptions
made
i
your
answer.
If
no
assumptions
are
needed.
you
should
state
this
explicitly.
Section
B
(34
marks)
Answer
this
section
in
a
separate
answer
booklet.
Juestion
4
'onsider
an
insurance
sold
to
a
life
aged exactly
x.
The
insurance
pays
$f
at
the
end
of
the
year
of
death
if
the
life
dies
between
ages
x+
;
exact
and
x+k
exact.
The
insurance
also
pays
$g
either
on
survival
to
age
x+m
exact
or
at
the
end
of
the
year
of
death
if
the
life
dies
between
ages
x
+/
exact
and
x+
m
exact.
Note
that
x,
j,k,I,m
are
all
positive
integers
and
that
j
<k
</
<m.
You
are
also
given
that
[
#
g.
[3]
(a)
Write
down
an
expression
for
the
present
value
of
the
benefits
paid
under
this
insurance
in
terms
of
a
suitable
lifetime
random
variable.
[3]
(b)
Write
down
an
expression,
using
actuarial
notation,
for
the
expected
present
value
of
the
benefits
paid
under
this
insurance.
[3]
(¢)
Suppose
that
mortality
follows
AM92
Ultimate
and
that
the
interest
rate
is
4%
per
annum
effective.
Given
that
x=60,f=1,¢g=2,j=5k=10,/=15
and
m
=20,
calculate
the
expected
present
value
from
part
(b).
d)
Suppose
only
for
this
part
that
the
insurance
has
no
survival
benefit.
The
death
benefits
described
above
continue
to
apply.
Find
an
expression,
using
actuarial
notation
for
the
variance
of
the
present
value
of
the
benefits
paid
under
this
insurance.
Give
your
answer
in
terms
of
x,
f,
g,
j,k,/
and
m.
Do
NOT
use
numerical
values
from
part
(c).