Answer the following microeconomics questions well with explanation

16.1 If Ty and Ky are random variables measuring the complete and curtate future lifetimes, respectively, of a life aged x , write down an expression for each of the following symbols as the expectation of a random variable: Ax (ii) (iii) enl 16.2 The benefit payable under a special assurance policy has present value random variable: W = vmax(K*+1, n) where Kx is the curtate future lifetime of a person currently aged exactly x. (i) Describe the benefit paid under this policy. (ii) Express [W ] in terms of standard actuarial notation. (iii) Express var[W] in terms of standard actuarial notation. 16.3 Calculate the expectation and standard deviation of the present value of the benefits from each of the following contracts issued to a life aged exactly 40, assuming that the annual effective interest rate is 4% and AM92 Ultimate mortality applies: (i) a 20-year pure endowment, with a benefit of $10,000 a deferred whole life assurance with a deferred period of 20 years, under which the death benefit of f20,000 is paid at the end of the year of death, as long as this occurs after the deferred period has elapsed. 16.4 Using an interest rate of 6% pa effective and AM92 Ultimate mortality, calculate: (i) 1 50:15 (ii) Aso:151 16.5 If /40 = 1,000 and /40+t =/40 -5t for t=1,2,..,10, calculate the value of Aqp:10) at 6% pa interest.16.6 A life insurance company issues a 3-year term assurance contract to a life aged exactly 42. The sum assured of 10,000 is payable at the end of the policy year of death. Calculate the expected present value of these benefits assuming AM92 Select mortality and an interest rate of 5% pa effective. 16.7 A whole life assurance policy pays $10,000 immediately on death of a policyholder currently aged 50 exact, but only if death occurs after the age of 60. (i) Write down an expression for the present value random variable of the benefit payable under this policy. (ii) Determine an expression, in the form of an integral, for the expected present value of the benefit payment, and express your answer using standard actuarial notation. (iii) Determine an expression for the variance of the present value of the benefit payment, expressing your answer using standard actuarial notation