Answer the following microeconomics questions clearly with explanation

Example 6.8 Calculate the gross single premium for a deferred annuity of $80 000 per year payable monthly in advance, issued to a select life now aged 50 with the first annuity payment on the life's 65th birthday. Allow for initial expenses of $1 000, and renewal expenses on each anniversary of the issue date, provided that the policyholder is alive. Assume that the renewal expense will be $20 on the first anniversary of the issue date, and that expenses will increase with inflation from that date at the compound rate of 1% per year. Assume the Standard Select Survival Model with interest at 5% per year.l. 1" Example 6.9 A life insurer is about to issue a 25-year endowment insurance with a basic sum insured of $251} [] to a select life aged exactly 31]. Premiums are payable mutually throughout the term of the policy. Initial expenses are $121)!) plus 441% of the rst premium and renewal expenses are 1% of the second and subsequent premiums. The insurer allows for a compon mversiary bonus of 2.5% of the basic sum insured, vesting on each policy anniversary {including the last). The death benet is payable at the end of the year of death. Assume the Standard Select Survival Model with interest at 5% per year. (a) Derive an expression for the future loss random variable, LE , for this policy. (b) Calculate the annual premium for this policy. (c) Let 143(k) denote the present value of the loss on the policy given that Kl = k for k 5 24 and let Lg{25) denote the present value of the loss on the policy given that the policyholder survives to age 55. Calculate 143(k) fork =,1,...,25. (d) Calculate the probability that the insurer makes a prot on this policy. (e) Calculate tang]. Example 6.10 An insurance company is about to issue a single premium deferred annuity to a select life aged 55. The first annuity payment will take place 10 years from issue, and payments will be annual. The first annuity pay- ment will be $50 000, and each subsequent payment will be 3% greater than the previous payment. Ignoring expenses, and using the Standard Select Survival Model with interest at 5% per year, calculate (a) the single premium, (b) the probability the insurance company makes a profit from this policy, and (c) the probability that the present value of the loss exceeds $100 000