You are given the following Markov chain, representing a multiple lives model:

You are also given, for $t>0$ :

${}_{x+t\text{:}y+t}^{01}=0\mathrm{.}02$

${}_{x+t\text{:}y+t}^{02}=0\mathrm{.}04$

${}_{x+t}^{13}=0\mathrm{.}10$

${}_{y+t}^{23}=0\mathrm{.}03$

Calculate the probability, given that both are alive at time $0,$ that neither will be alive at time

For annual premium unit whole life insurances on two independent lives $(x)$ and $($y$),$

with benefits payable at the end of the year of death, you are given

The annual net premium for an insurance on $(x)$ is $0\mathrm{.}10.$

The annual net premium for an insurance on $(y)$ is $0\mathrm{.}14.$

The annual net premium for an insurance on $(xy)$ is $0\mathrm{.}19.$

$d=0\mathrm{.}06$

Calculate the annual net premium for an insurance on the last survivor of $(x)$ and $(y).$

For two independent lives $(x)$ and $(y),$ you are given:

${}_x=0\mathrm{.}04$

${}_y=0\mathrm{.}02$

$=0\mathrm{.}08$

A fully continuous insurance pays $1000$ on the first death if it occurs within $10$ years

and $2000$ on the second death if it occurs within $20$ years. Premiums are payable while

both are alive, but for no longer than $20$ years. Calculate the level annual net premium.