I need help on approaching a particular question regarding actuarial science. I just need a rough concept on how to do it, so I won't provide any extra data to keep this simple.
For these questions, I have built a cohort life expectancy data which shows px, qx, ex, etc. (Everything you will see in a life table), starting from 1930 until 2020.
Say there are two siblings, a brother and a sister. Now at year 1980, the brother is aged 50 and the sister is aged 40. What is the probability that they both live until age 70? Do I just multiply the probability of each person to lived until age 70 together?
Then another question, in year 1980, the brother also bought a life insurance which would pay $30000 if he were to die before the age of 75 if the sister were still alive at that time. Based on the cohort data that I have computed, what would be the actuarial present value of this insurance in 1980?
For $(40)$ and $(50)$ to both live to age $70$, the probability is simply $$({}_{30} p_{40})({}_{20} p_{50})$$ unless you have different tables for males and females, or you do not assume that family members have independent future lifetimes.
For the second question, the actuarial present value of a term life insurance of $n$ years payable upon death of $(x)$ before age $x+n$ as long as $(y)$ is alive, is expressed as $$\overline{A}_{{\overset{1}{x}y:\overline{n}\rceil}} = \int_{t=0}^n v^t {}_t p_{xy} \mu(x+t) \, dt$$ where ${}_t p_{xy}$ is the joint survivorship function. In the case where the lives are independent, ${}_t p_{xy} = ({}_t p_x)({}_t p_y)$.