CHAPTER 15 THE EFFECTS OF TIME AND RISK ON VALUE Test Problems How much will a $50 deposit made today be worth in 20 years if the compound rate of interest is 10 percent? d. $336.37 How much would you pay for the right to receive $80 at the end of 10 years if you can earn 15 percent interest? b. $19.77 How much would you pay to receive $50 in one year and $60 in the second year if you can earn 15 percent interest? a. $88.85 What amount invested at the end of each year at 10 percent annually will grow to $10,000 at the end of five years? b. $1,637.97 How much would you pay for the right to receive nothing a year for the next 10 years and $300 a year for the following 10 years if you can earn 15 percent interest? a. $372.17 What is the present value of $500 received at the end of each of the next three years and $1,000 received at the end of the fourth year, assuming a required rate of return of 15 percent? c. $1,713.37 If a landowner purchased a vacant lot six years ago for $25,000, assuming no income or holding costs during the interim period, what price would the landowner need to receive today to yield a 10 percent annual return? c. $44,289.03 What is the present value of the following series of cash flows discounted at 12 percent: $40,000 now; $50,000 at the end of the first year; $0 at the end of year the second year; $60,000 at the end of the third year; and $70,000 at the end of the fourth year? d. $171,835.94 Assume an income-producing property is priced at $5,000 and has the following income stream (year 1, $1,000; year 2, -$2,000; year 3, $3,000; and year 4, $3,000). Would an investor with a required rate of return of 15 percent be wise to invest at the current price? b. No, because the project has a net present value of -$1,954.91. Study Questions 1. Dr. Bob Jackson owns a parcel of land that a local farmer has offered to rent for the next 10 years. The farmer has offered to pay $20,000 today or an annuity of $3,200 at the end of each of the next 10 years. Which payment method should Dr. Jackson accept if his required rate of return is 10 percent? Solution: Dr. Jackson should choose the payment method that maximizes his net present value. If he chooses the lump sum payment, the net present value is simply the $20,000 he will receive today. If he chooses the annuity plan, the net present value will be only $19,662.61. N = 10 I = 10 % PV =? PMT = 3,200 FV = 0 Therefore, Dr. Jackson should choose the lump sum payment of $20,000. 2. You are able to buy an investment for $1,000 that gives you the right to receive $438 in each of the next three years. What is the internal rate of return on this investment? Solution: This is simply a yield calculation problem. Like any time-value-of-money problem, we are given four inputs and are asked to solve for the fifth. In this case, we must solve for the interest rate as follows: N = 3 I =? PV = -1,000 PMT = 438 FV = 0 Solving this setup tells us the above loan yields a 15 percent return. 3. Calculate the present value of the income stream given below assuming discount rates of 8 percent and 20 percent. Year Income 1 $3,000 2 $4,000 3 $6,000 4 $1,000 Solution: This problem is solved by entering the annual income stream and discount rate into the cash flow registers of any standard financial calculator and solving for the net present value. Assuming an 8% discount rate, the income stream is valued at $11,705.16. Alternatively, if the discount rate is 20%, the value of the income stream will be $9,232.25. 4. Calculate the IRR and NPV for the following investment opportunities. Assume a 16 percent discount rate for the NPV calculations: Year Project 1 Cash Flow Project 2 Cash Flow 0 -$10,000 -$10,000 1 1,000 1,000 2 2,000 12,000 3 12,000 1,800 Solution: To solve this problem, simply enter each set of cash flows into the cash flow registers of your financial calculator and ask it to find the IRR. For Project 1, the internal rate of return is 16.16%, while for Project 2, the internal rate of return is 21.23%. The NPV for Project 1 is $36.29 and the NPV for Project 2 is $933.21. If these projects were independent, each IRR should be individually compared to the required rate of return to determine whether the investment should be made. However, if the projects are mutually exclusive and are of equivalent risk, Project 2 is preferred to Project 1. Addtionally, the higher NPV of Project 2 clearly makes this alternative the most attractive investment option because the investorâ€™s net worth will increase by $933.21. 5. How much would you pay for an investment that provides $1,000 at the end of the first year if your required rate of return is 10 percent? Now compute how much you would pay at 8 percent and 12 percent rates of return. Solution: At 10%, an investor would be willing to pay $909.09. N = 1 I = 10 PV = ? PMT = 0 FV = 1,000 At 8%, an investor would be willing to pay $925.93. N = 1 I = 8 PV = ? PMT = 0 FV = 1,000 At 12%, an investor would be willing to pay $892.86. N = 1 I = 12 PV = ? PMT = 0 FV = 1,000 6. Your grandmother gives you $10,000 to be invested in one of three opportunities: real estate, bonds, or zero coupon bonds. If you invest the entire $10,000 in one of these opportunities with the cash flows shown below, which investment offers the highest NPV? Assume an 11 percent discount rate is appropriate for all three investments Year 1 Year 2 Year 3 Year 4 Year 5 Real Estate $1,300 $1,300 $1,300 $1,300 $9,000 Bond $1,000 $1,000 $1,000 $1,000 $11,000 Zero Coupon $0 $0 $0 $0 $18,000 Solution: Entering the annual income stream and discount rate into the cash flow registers of our financial calculator, we obtain the following net present value calculations: real estate, (625.76); bond, (369.5); and zero coupon, 682.12. 7. If you purchase a parcel of land today for $25,000 and you expect it to appreciate 10 percent per year in value, how much will your land be worth 10 years from now? Solution: At a 10% discount rate, the investment will be worth $64,843.56 in ten years. N = 10 I = 10 PV = -25,000 PMT = 0 FV = ? 8. If you deposit $1 at the end of each of the next ten years and these deposits earn interest at 10 percent, what will the series of deposits be worth at the end of the 10th year? Solution: At a 10% discount rate, this series of payments, or annuity, will be worth $15.94 in ten years. N = 10 I = 10 PV = 0 PMT = 1 FV = ? 9. If you deposit $50 per month in a savings and loan association at 10 percent interest, how much will you have in your account at the end of the 12th year? Solution: At a 10% discount rate, this series of payments, or annuity, will be worth $13,821.89 in 12 years. N = 144 I = 10/12 PV = 0 PMT = 50 FV = ? 10. If your parents purchased an endowment policy of $10,000 for you and the policy will mature in 12 years, how much is it worth today, discounted at 15 percent? Solution: At a 15% discount rate, the present value of this future payment is $1,869.07. N = 12 I = 15 PV = ? PMT = 0 FV = 10,000 11. A family trust will convey property to you in 15 years. If the property is expected to be worth $50,000 when you receive it, what is the present value of your interest, discounted at 10 percent? Solution: At a 10% discount rate, the present value of this future payment is $11,969.60. N = 15 I = 10 PV = ? PMT = 0 FV = 50,000 12. You want to buy a house for which the owner is asking $625,000. The only problem is that the house is leased to someone else with five years remaining on the lease. However, you like the house and believe it will be a good investment. How much should you pay for the house today if you could strike a bargain with the owner under which she would continue receiving all rental payments until the end of the leasehold at which time you would obtain title and possession of the property? You believe the property will be worth the same in five years as it is worth today and that this future value should be discounted at a 10 percent annual rate. Solution: This problem requires you to determine the present value of the house today if you are willing to purchase it for $625,000 five years from today. Using a 10% discount rate, the home is worth $388,075.83 today. N = 5 I = 10 PV = ? PMT = 0 FV = 625,000 13. If someone pays you $1 a year for 20 years, what is the value of the series of future payments discounted at 10 percent annually? Solution: At a 10% discount rate, the present value of this series of future payments, or annuity, is $8.51. N = 20 I = 10 PV = ? PMT = 1 FV = 0 14. You are at retirement age and one of your benefit options is to accept an annual annuity of $7,500 for 15 years. What lump sum settlement, if paid today, would have the same present value as the $7,500 annual annuity? Assume a 10 percent discount rate. Solution: At a 10% discount rate, the present value of this series of future payments is $57,045.60. This is the lump sum equivalent of receiving $7,500 for 15 years. N = 15 I = 10 PV = ? PMT = 7,500 FV = 0 15. What monthly deposit is required to accumulate $10,000 in eight years if the deposits are compounded at an annual rate of 8 percent? Solution: Assuming an 8% discount rate and a future value of $10,000, the monthly amount required to be deposited is $74.70. N = 96 I = 8/12 PV = 0 PMT = ? FV = $10,000 16. You are thinking about purchasing some vacant land. You expect to be able to sell the land ten years from now for $500,000. What is the most you can pay for the land today if your required rate of return is 15 percent? What is the expected (annualized) return on this investment over the 10-year holding period if you purchase the land for $170,000? Solution: The maximum amount you can spend to purchase this property is the present value of the future price, discounted at 15 percent for ten years. Using a financial calculator, this amount is $123,592.35. N = 10 I = 15 PV = ? PMT = 0 FV = $500,000 The expected annualized return on this investment can be solved using a financial to obtain for the interest rate that equates a present value of $170,000 to $500,000 in ten years. The annualized return of this investment is 11.39% N = 10 I = ? PV = -170,000 PMT = 0 FV = $500,000 Alternatively, the cash flow function can be used to calculate the IRR of this investment, whereby the initial cash outflow at time zero is $170,000, the cash flows for the time period 1-9 is zero, and the cash flow received in year 10 is $500,000. Using this approach, the IRR is 11.39%. 17. You are considering the purchase of a small income-producing property for $150,000 that is expected to produce the following net cash flows: End of Year Cash Flow $50,000 $50,000 $50,000 $50,000 Assume your required internal rate of return on similar investments is 11 percent. What is the net present value of this investment opportunity? What is the going-in internal rate of return on this investment? Solution: Using the cash flow function on a financial calculator and entering the information provided above, the NPV of this investment is $5,122.28. Alternatively, the NPV can be solved as follows: N = 4 I = 11 % PV = ? PMT = $50,000 FV = 0 The present value of this series of payments is $155,122.28. Subtracting the amount of the cash outflow at period zero ($150,000), the present value is also $5,122.28. The going-in IRR for this investment is 12.59%. PAGE 15- PAGE 1