CHAPTER 14
The Effects of Time and Risk on Value
Test Problems
1. How much will a $50 deposit made today be worth in 20 years if interest is
compounded annually at a rate of 10 percent?
2. How much would you pay today for the right to receive $80 at the end of 10 years
if you can earn 15 percent interest on alternative investments of similar risk?
3. How much would you pay today to receive $50 in one year and $60 in the second
year if you can earn 15 percent interest on alternative investments of similar risk?
4. What amount invested at the end of each year at 10 percent annually will grow to
$10,000 at the end of five years?
5. How much would you pay today for the right to receive nothing for the next 10
years and $300 a year for the following 10 years if you can earn 15 percent
interest on alternative investments of similar risk?
6. 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?
7. 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 on the land
investment?
8. 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?
9. Assume an investment 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 a price of
$5,000?
10. As the level of perceived risk increases,
Study Questions
1. Dr. Bob Jackson owns a parcel of land that a local farmer has offered to rent from Dr.
Bob 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
N = 10 I = 10 % PV =? PMT = 3,200 FV = 0
2. You are able to buy an investment today 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
must solve for the interest rate as follows:
N = 3 I =? PV = -1,000 PMT = 438 FV = 0
3. Calculate the present value of the income stream given below assuming a discount rate
of 8 percent. What happens to present value if the discount rate increases to 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
Alternatively, if the discount rate is 20%, the value of the income stream will be
4. Calculate the IRR and NPV for the following two 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
5. How much would you pay today 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
N = 1 I = 8 PV = ? PMT = 0 FV = 1,000
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, regular bonds, or zero coupon bonds. If you invest the entire $10,000 in one of
these opportunities with the expected cash flows shown below, which investment offers
the highest NPV? Assume for simplicity that 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
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
assuming annual compounding?
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 compounded annually, 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
N = 10 I = 10 PV = 0 PMT = 1 FV = ?
9. If you deposit $50 per month in a bank account at 10 percent annual interest
(compounded monthly), 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
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 annually?
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 annually?
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 five-year 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
N = 5 I = 10 PV = ? PMT = 0 FV = 625,000
13. If someone pays you $1 a year for 20 years, what is the present 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
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 $75,000 for 15 years. The first payment would be received one year from
today. What lump sum settlement, if paid today, would have the same present value as the
$75,000 annual annuity? Assume a 10 percent annual discount rate.
Solution: At a 10% discount rate, the present value of this series of future payments is
N = 15 I = 10 PV = ? PMT = 75,000 FV = 0
15. What monthly deposit is required to accumulate $10,000 in eight years if the deposits
earn an annual rate of 8 percent, compounded monthly?
Solution: Assuming an 8% discount rate and a future value of $10,000, the monthly
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
N = 10 I = 15 PV = ? PMT = 0 FV = $500,000
The expected annualized return on this investment can be solved using a financial to
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
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
1 $50,000
2 $50,000
3 $50,000
4 $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? Should you make the investment?
Solution: Using the cash flow function on a financial calculator and entering 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 going-in IRR for this investment is 12.59%. Yes, you should make the investment.
18. Raw land at the edge of urban development that lacks the necessary permits for
development is one of the most risky kinds of real estate investment. Defend or refute this
assertion.
Solution: Evaluated against the two types of investment risk confronting real estate
investors, uncertainty of costs and uncertainty of value, raw land lacking permitting can
be viewed as the riskiest form of real estate investment. Raw land at the edge of urban
19. You are contemplating replacing your conventional hot water heater with a solar hot
water heater system at a cost of $4,000. How should you define the potential benefits
that you need to receive to justify the investment?
Solution: The potential benefit gained from this investment is a reduction in future utility
costs. This purchase requires an analysis of the initial costs and the value of the future
20. Solve for the unknown discount rate in each of the following:
Present Value Years Discount Rate Future Value
$2,400 2 11.24% $2,970
3,600 10 11.61 10,800
21. Solve for the unknown number of years in each of the following:
Present Value Years Discount Rate Future Value
$5,600 9.63 9% $12,840
8,100 17.61 10 43,410
22. Assume the total cost of a college education will be $310,000 when your infant child
enters college in 18 years. How much you invest at the end of each month in order to
accumulate the required $310,000 at the end of 18 years if your monthly investments earn
an annual interest rate of 5 percent, compounded monthly?
N = 18 x 12 I/YR = 5/12 PV = 0 PMT = ? FV = 310,000
23. You are trying to accumulate a $40,000 down payment to purchase a home. You can
afford to save $1,000 per quarter. If these quarterly investments earn an annual rate of 7
percent, how many quarters will it take to reach your goal?
N = ? I/YR = 7/4 PV = 0 PMT = -1,000 FV = 40,000
24. You have signed a new lease today to rent office space for five years. The lease
payments are fixed at $4,500 per month for the first two years, but rise to $5,500 per
month in years 3-5. What is the present value of this lease obligation if the appropriate
discount rate is 8 percent?
Present value of $5,500 per month for five years:
N = 60 I/YR = 8/12 PV = ? PMT = 5,500 FV = 0
Present value of $1,000 per month for two years:
N = 24 I/YR = 8/12 PV = ? PMT = 1,000 FV = 0
25. Suppose you are going to receive $10,000 per year for five years. The appropriate
interest/discount rate is 11 percent.
a. What is the present value of the payments if they are in the form of an ordinary
annuity?
N = 5 I/YR = 11 PV = ? PMT = 10,000 FV = 0
What is the present value if the payments are an annuity due?
b. Suppose you plan to invest the payments for five years. What is the future value if
the payments are an ordinary annuity?
N = 5 I/YR = 11 PV = 0 PMT = 10,000 FV = ?
What if the payments are an annuity-due?