Answer all the questions correctly with well explanation
17.1 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: (i) (tb) (iii) x:n 17.2 Calculate the expectation of the present value of the benefits from each of the following contracts issued to a life aged exactly 45, assuming that the annual effective interest rate is 4%, and AM92 Select mortality applies: (i) a deferred whole life annuity-due, with a deferred period of 15 years, under which payments of $5,000 are made annually in advance while the policyholder is alive after the deferred period has elapsed (ii) a guaranteed annuity, with a guarantee period of 15 years, under which payments of E5,000 are made annually in arrears for a minimum of 15 years and for life thereafter. 17.3 (1) Let Z be a random variable representing the present value of the benefits payable under m style an immediate life annuity that pays 1 per year in advance, issued to a life aged x. Show that var(Z) = 2 (2Ax -(Ax)? ), where 2 Ax is an assurance calculated at a rate of interest which you should specify. [4] (ii) A life office issues such a policy to a life aged exactly 65. The benefit is $275 per annum. Calculate the standard deviation of the annuity. Basis: Mortality: AM92 Ultimate Interest: 6% per annum throughout [3] [Total 7]17.4 A special 25-year life insurance policy is issued to a life aged x and provides the following benefits: a lump sum of $75,000 (payable at the end of the policy year) if death occurs during the first 10 years a dependants' pension (payable in the form of an annuity certain) of $5,000 pa payable on each remaining policy anniversary during the term (including the 25th anniversary) if death occurs after 10 years but before the end of the term of the policy a pension of $7,500 pa commencing on the day after the term of the policy expires and with payments on each subsequent policy anniversary while the policyholder is still alive. Write down an expression for the present value random variable of the benefits under this policy. 17.5 A life currently aged x is subject to a constant force of mortality of 0.02 pa. The constant force of interest is 0.03 pa. Calculate: (Wi) 17.6 An annuity is payable continuously throughout the lifetime of a person now aged exactly 60, but m style for at most 10 years. The rate of payment at all times t during the first 5 years is f10,000 pa, and thereafter it is f12,000 pa. The force of mortality of this life is 0.03 pa between the ages of 60 and 65, and 0.04 pa between the ages of 65 and 70. Calculate the expected present value of this annuity assuming a force of interest of 0.05 pa. [5]
