Solving for Variables Other Than Present Value or Future Value

29. You have decided to buy a perpetuity. The bond makes one payment at the end of every year forever and has an interest rate of 5%. If you initially put $1000 into the bond, what is the payment every year?

30. You are thinking of purchasing a house. The house costs $350,000. You have $50,000 in cash that you can use as a down payment on the house, but you need to borrow the rest of the purchase price. The bank is offering a 30-year mortgage that requires annual payments and has an interest rate of 7% per year. What will your annual payment be if you sign up for this mortgage?

* 31. You are thinking about buying a piece of art that costs $50,000. The art dealer is proposing the following deal: He will lend you the money, and you will repay the loan by making the same payment every two years for the next 20 years (i.e., a total of 10 payments). If the interest rate is 4% per year, how much will you have to pay every two years?

*32. You would like to buy the house and take the mortgage described in Problem 30. You can afford to pay only $23,500 per year. The bank agrees to allow you to pay this amount each year, yet still borrow $300,000. At the end of the mortgage (in 30 years), you must make a balloon payment; that is, you must repay the remaining balance on the mortgage. How much will this balloon payment be?

*33. You are saving for retirement. To live comfortably, you decide you will need to save $2 million by the time you are 65. Today is your twenty-second birthday, and you decide, starting today and continuing on every birthday up to and including your sixty-fifth birthday, that you will put the same amount into a savings account. If the interest rate is 5%, how much must you set aside each year to make sure that you will have $2 million in the account on your sixty-fifth birthday?

* 34. You realize that the plan in Problem 33 has a flaw. Because your income will increase over your lifetime, it would be more realistic to save less now and more later. Instead of putting the same amount aside each year, you decide to let the amount that you set aside grow by 7% per year. Under this plan, how much will you put into the account today? (Recall that you are planning to make the first contribution to the account today.)

35. You have an investment opportunity that requires an initial investment of $5000 today and will pay $6000 in one year. What is the IRR of this opportunity?

36. You are shopping for a car and read the following advertisement in the newspaper: "Own a new Spitfire! No money down. Four annual payments of just $10,000." You have shopped around and know that you can buy a Spitfire for cash for $32,500. What is the interest rate the dealer is advertising (what is the IRR of the loan in the advertisement)? Assume that you must make the annual payments at the end of each year.

37. A local bank is running the following advertisement in the newspaper: "For just $1000 we will pay you $100 forever!" The fine print in the ad says that for a $1000 deposit, the bank will pay $100 every year in perpetuity, starting one year after the deposit is made. What interest rate is the bank advertising (what is the IRR of this investment)?

*39. Your grandmother bought an annuity from Great-West Life Insurance Company for $200,000 when she retired. In exchange for the $200,000, Great-West Life will pay her $25,000 per year until she dies. The interest rate is 5%. How long must she live after the day she retired to come out ahead (that is, to get more in value than what she paid in)?

*40. You are thinking of making an investment in a new plant. The plant will generate revenues of $1 million per year for as long as you maintain it. You expect that the maintenance costs will start at $50,000 per year and will increase 5% per year thereafter. Assume that all revenue and maintenance costs occur at the end of the year. You intend to run the plant as long as it continues to make a positive cash flow (as long as the cash generated by the plant exceeds the maintenance costs). The plant can be built and become operational immediately. If the plant costs $10 million to build, and the interest rate is 6% per year, should you invest in the plant?

*41. You have just turned 22 years old, have just received your bachelor's degree, and have accepted your first job. Now you must decide how much money to put into your registered retirement savings plan (RRSP). The RRSP works as follows: Every dollar in the RRSP earns 7% per year. You will not make withdrawals until you retire on your sixty-fifth birthday. After that point, you can make withdrawals as you see fit. You decide that you will plan to live to 100 and work until you turn 65. You estimate that to live comfortably in retirement, you will need $100,000 per year, starting at the end of the first year of retirement (i.e., when you turn 66) and ending on your one-hundredth birthday. You will contribute the same amount to the RRSP at the end of every year that you work. How much do you need to contribute each year to fund your retirement?

*42. Problem 41 is not very realistic because most people do not contribute a fixed amount to their RRSP each year. Instead, they are more likely to contribute a fixed percentage of their salary. Assume that your starting salary is $45,000 per year and it will grow 3% per year until you retire. Assuming everything else stays the same as in Problem 41, what percentage of your income do you need to contribute to the RRSP every year to fund the same retirement income?