$a[6$ marks$]$ A male life aged $55$ exact and a female life aged $53$ exact purchased a $20-$year joint life term assurance. The policy pays a sum assured of e$150,000$ payable at the end of the year of the first death if it occurs within the policy term. Calculate the expected present value of the death benefit given that the interest rate is $4\%$ per annum and from the PMA$92$C$20$ and PFA$92$C$20$ tables note a $55$:$53=16\mathrm{.}284,$ a $75$:$73=7\mathrm{.}975$ and $v^{20}$p$55$:$53=0\mathrm{.}353725.$ b$)$ A life insurance company issues a reversionary annuity policy to a male aged $55$ exact and a female aged $53$ exact. The annuity of e$70,000$ per annum payable monthly in arrears com$-$ mences if the male dies first and payments cease on the death of the female or on the $20$th anniversary of the policy inception if ear$-$ lier. Calculate $($with the usual assumptions$)$ the single premium of the policy given that the interest rate is $4\%$ per annum and note the information in $($a$)$ and also from the PFA$92$C$20$ tables note a $53=18\mathrm{.}768,$ a $73=11\mathrm{.}734$ and $20$p$53=0\mathrm{.}91334.$