2 Annuity (50') In this exercise, we analyze the HKMC Annuity Plan (simplified). The following table lists infor- mation of this plan. For more details, check this website. In this exercise, we assume the buyer is male. The first column lists the age of the insured at the purchase. The second column lists the guaranteed monthly annuity payment. The third and fourth columns list the number of guaran- teed period of annuity payments, which we will analyze further in subquestions. For simplicity, we assume that payments are made annually. For example, if a 65-year-old buyer buys the annuity, he gets an annual payment of 5800 12 at the end of every year if he survives. For simplicity, you don't need to calculate the payment and death probability in each month of the year. The pay- ment ends when the insured is 100 years old. Use the Life Table attached to answer the following questions. Age Last Birthday at Application Male (Single Premium HK$1,000,000) Guaranteed Monthly Annuity Payment (The amount will be determined at the outset and will remain unchanged throughout the whole of your life) Number of Guaranteed Monthly Annuity Payments Guaranteed Period 232 233 23 60 5,100 206 206 months 61 5,240 201 201 months 62 5,380 196 196 months 63 5,520 191 191 months 64 5,660 186 186 months 65 5,800 182 182 months 66 5,950 177 177 months 67 6,100 173 173 months 68 6,250 168 168 months 69 6,400 165 165 months 70 6,560 161 161 months 71 6,720 157 157 months 72 73 223 6,880 153 153 months 7,040 150 150 months 74 7,200 146 146 months 75 7,360 143 143 months 167 7,520 140 140 months 77 7,680 137 137 months 78 7,840 134 134 months 79 8,000 132 132 months 80 8,160 129 129 months 81 8,320 127 127 months 82 8,480 124 124 months 83 8,640 122 122 months 84 8,800 120 120 months 85 8,960 118 118 months 1. Suppose the policy does not have any death benefit, which means if the insured dies the policy stops making payments (i.e., ignore Column 3 and 4). Assume the policyholder does not surrender the policy before death. (10') (a) Calculate the internal rate of return (IRR) of this policy for buyers of 65 (the buyer is right before his 65th birthday). Assume the first payment starts one year later. That is, the buyer can receive 5800 12 at the end of every year. Note: the IRR is such that the PV of premium equals the PV of expected benefits. (8') (b) Suppose there is another financial instrument (e.g., a bond) that can achieve the same level of IRR as you calculate in part (a). What is the benefit of the annuity plan, compared with the alternative instrument? (2') 2. Death benefit and IRR. (20') The death benefit we consider is simpler than the death benefit from the actual plan (if you read the document through the link). Assume that if the insured dies within the guaranteed period, the beneficiary can receive the same amount of income for several months (the number of months with guaranteed income is specified in the last column for different ages). (a) Recalculate your answer to question 1(a). For simplicity, the guaranteed 182 months can be approximated by 180 months (15 years). (10') (b) Calculate the IRR for a buyer of 80 years old. For simplicity, the guaranteed 129 months can be approxiated by 132 months (11 years). (8%) (c) Compare the two IRRs. Do older buyers get a higher or lower IRR than younger buyers? What are possible reasons? (2') 3. Policy surrender. (20') In this subquestion, we examine the surrender of the policy. Suppose a buyer purchases the policy right after his 65th birthday and receives the specified payments until right before his 75th birthday. Then he unfortunately is diagnosed with cancer and needs to pay a large amount of medical expenses (0.7 million HKD). (a) Suppose the annual interest credited to the account is 5%, and assume the premium loading is 0. Assume there exists a death benefit: the income payment will continue for at least 182 months (you may use 180 months, or 15 years as an approximation) if the annuitant dies before 80. Calculate the theoretical surrender value of the policy when the buyer surrenders the policy if there is no premium loading. Does the theoretical surrender value increase or decrease with age? Why? (10') (b) Explain why such policy surrender brings adverse selection problem to the annuity policy. (5') (c) If you are a financial consultant to an elderly who is going to enter into the annuity policy, can you provide a reason to convince the buyer to simultaneously buy a health insurance? (5')