Respond to the following questions appropriately.

A 20-year with-profits endowment with a basic sum assured of $15,000 was issued to a man aged 45 exact at entry. The policy has been in force for 7 years. The policy is subject to simple reversionary bonuses that vest at the start of each year, and death benefits are paid at the end of the policy year of death. In addition, a terminal bonus equal to 25% of the sum assured plus existing bonus, is payable on a death or maturity claim. Simple bonuses were declared at the rate of 4% for the first year of the policy, and 3.75% pa thereafter to date. The company expects to pay future bonuses at a rate of 3.5% pa from now on. Calculate the net premium reserve for this policy as at the present time. Assume AM92 Ultimate mortality and 4% pa interest.1. Consider the following health status problem: max y = - (21 - 1)2 - (x2 -2)2 + 10, subject to : ntrz=1, where y represents an individual's health, and r, and r, are daily dosages of two health-enhencing drugs. (a) If there is no constraint (2), find optimal (21, 12) and the corresponding maximum value of y-(b) In the presence of constraint (2), find optimal (21, 12, A) (X is the La- grangian multiplier for this constrained optimization problem) and the corre- sponding maximum value of y, and show that A is equal to the reduction of the maximum value of y due to the constraint ri + 12 = 1