5. (30 points) You are planning to save towards your retirement with increasing monthly payments (first payment is $2,000, and monthly growth rate is 0.1%) for 40 years. The first payment will come in one month. You save into a retirement account paying an APR of 12%, compounded monthly. You plan to retire immediately after the last payment, and then, starting from the following month, withdraw a constant amount of money from the account, until it is empty. The rate on your account is still 12% APR, compounded monthly. You believe you are going to live 30 years after retiring. (a) (8 points) How much can you withdraw from your account every month during the retirement? (b) (7 points) Now suppose that you want to leave your kids an inheritance of $10,000,000. That is, you still want to withdraw a constant amount of money from your account every month for 30 years. However, you want also to have $10,000,000 in your account by the end of year 30. Suppose that your savings for the retirement do not change. What is a new regular monthly payment of the retirement account? (c) (7 points) Suppose now that you decided not to leave any inheritance (so that you follow the repayment scheme of part a). However, at the end of year 20 of your retirement you suddenly realize that you are likely to live for 20 years more (instead of 10 years as you initially expected). You still want to get fixed monthly payments from your account. What are these payments during the last 20 years of your retirement? (d) (8 points) Suppose again that you do not want to leave any inheritance and you expect to live for 30 years after retirement. After 35 years of work, your pension fund proposes you a deal: instead of following your initial plan, you can get a one-time lump sum payment at the moment of retirement (which happens 5 years after this proposal). If you take their proposal, you do not need to make any additional payments to the retirement fund; however, all your previous payments are going to be lost. What is the minimum lump sum payment that you will be willing to accept?