Question

Problem 7. (10 pts) The insurance company has a cohort of policyholders, 70% of whom are non-smokers and 30% of whom are smokers. For the fixed age x, the insurance company's model for mortality has ONS = ONS = 0.15 (non-smoker) and q = = 0.25 (smoker). (a). A policyholder is chosen at random from the cohort. Compute 19 and qx+1 for this poli- cyholder (b). Suppose that both non-smoker and smoker mortality follow UDD in each year of age. Compute the value Hx+0.2. Problem 8. (10 pts) Suppose that i = 0.04 and 25+1 = 0.02 for all t > 0. We know that Z1 is the discrete PVRV for a 20-year term endowment insurance on (25) and Z, is the discrete PVRV for a 20-year deferred, 10-year term insurance also on (25). Compute Var[Z1 + Z2]. Problem 9. (10 pts) You are given: (i) A = 0.35, (ii) 8 = 0.05 and (iii) Mate=0.04 for all t. Compute Anal

Answer 1

Answer:

(7).

Given that,

$\rightarrow$ The insurance company has a cohort of policyholders, 70% of whom are non-smokers and 30% of whom are smokers.

$\rightarrow$ For the fixed age x, the insurance company's model for mortality has qxNS =qx+1NS=0.15 (non-smoker) and qxS=qx+1S=0.25 (smoker).

(a).

$\rightarrow$ A policyholder is chosen at random from the cohort.

1qx=0.7$\times$ qxNS+0.3 $\times$ qxS [ I am assuming 1qx mean probability of policyholder dying within 1 year from age % I don't understand why they have mentioned 1 before q, if not necessary].

=0.7 $\times$ 0.15+0.3 $\times$ 0.25

=0.105+0.075

=0.18

q(x+1)=0.7 $\times$ qx+1NS+0.3 $\times$ qx+1S

=0.7 $\times$ 0.15+0.3 $\times$ 0.25

=0.18

(b).

$\rightarrow$ Suppose that both non-smoker and smoker mortality follow UDD in each year of age.

Under assumption of UDD.

* Th=5+en

Hence,
Mr+0.2
=0.18 | (1-0.2 $\times$ 0.18) |

=0.1867

(8).

$\rightarrow$ Suppose that i=0.04 and
J125++ = 0.02
for all t $\geq$ 0.

$\rightarrow$ We know that Z1 is the discrete PVRV for a 20-year term endownment insurance on (25) and Z2 is the discrete PVRV for a 20-year deferred, 10-year term insurance also on (25).

Z1= Vt , t< 20

=V25, t $\geq$ 20

Z2=0, t<20

= Vt , 20 $\leq$ t < 30

=0 ,   t $\geq$ 30

Z1+Z2 = Vt , t<20

= Vt +V25, 20 $\leq$ t < 30

=V25 , t $\geq$ 30

We can split this as

Y1=Vt , t<30

=0 , t $\geq$ 30

Y2=0 , t<20

=V25 ,t $\geq$ 20

Z1+Z2=Y1+ Y2

V(Z1+Z2)=V(Y1+ Y2)

=V(Y1) [Y2 is constant and Y1 is term assurance with term 30 years and age 25).

Now the variance of term assurance is given by, PAME(AM)? which can be found from actorial talxes.

(9).

Given ,

A1 1:ñ = 0.35
,
400 = 9
and
Mitt = 0.04
for all t.

EPV[ expected present value ] for term assurance=0.35.

EPV for endourment assurance =EPV for term assurance+Vnn$\times$ nPx

Where V=(1+i)(-1)=exp(-delta $\times$ n),nPx=exp(-uKn)

EPV for endourment assurance=0.35+e(-0.05 $\times$ n)  $\times$ e(-0.04$\times$n)

=0.35+e(-0.09 $\times$ n)

The final answer depends on n.