Answers to End-of-Chapter Problems B-49 Chapter 4: Net Present Value
Concept Questions - Chapter 4
4.1 Define future value and present value.
Future value is the value of a sum after investing over one or more periods.
Present value is the value today of cash flows to be received in the future.
How does one use net present value when making an investment decision?
One determines the present value of future cash flows and then subtracts the cost
of the investment. If this value is positive, the investment should be undertaken.
If the NPV is negative, then the investment should be rejected.
4.2 What is the difference between simple interest and compound interest?
With simple interest, the interest on the original investment is not reinvested.
With compound interest, each interest payment is reinvested and one earns
interest on interest.
What is the formula for the net present value of a project? T
NPV = -C0 + Ct /(1+I)t
t=1
4.3 What is a stated annual interest rate?
The stated annual interest rate is the annual interest rate without consideration of
compounding.
What is an effective annual interest rate?
An effective annual interest rate is a rate that takes compounding into account.
What is the relationship between the stated annual interest rate and the
effective annual interest rate?
Effective annual interest rate = (1 + (r/m) )m - 1.
Define continuous compounding.
Continuous compounding compounds investments every instant.
4.4 What are the formulas for perpetuity, growing-perpetuity, annuity, and
growing annuity?
Perpetuity: PV = C/r
Growing Perpetuity: PV = C/(r-g)
Annuity: PV = (C/r) [1-1/(1+r)T]
Answers to End-of-Chapter Problems B-50 Growing Annuity: PV = [C/(r-g)] [1-((1+g) / (1+r))T ]
What are three important points concerning the growing perpetuity formula?
1. The numerator.
2. The interest rate and the growth rate.
3. The timing assumption.
What are four tricks concerning annuities?
1. A delayed annuity.
2. An annuity in advance
3. An infrequent annuity
4. The equating of present values of two annuities.
Answers to End-of-Chapter Problems
Questions And Problems
Annual Compounding
4.1 Compute the future value of $1,000 compounded annually for
a. 10 years at 5 percent.
b. 10 years at 7 percent.
c. 20 years at 5 percent.
d. Why is the interest earned in part c not twice the amount earned in part a?
4.1 a. $1,000 1.0510 = $1,628.89
b. $1,000 1.0710 = $1,967.15
c. $1,000 1.0520 = $2,653.30
d. Interest compounds on the interest already earned. Therefore, the interest earned
in part c, $1,653.30, is more than double the amount earned in part a, $628.89.
4.2 Calculate the present value of the following cash flows discounted at 10 percent.
a. $1,000 received seven years from today.
b. $2,000 received one year from today.
c. $500 received eight years from today.
4.2 a. $1,000 / 1.17 = $513.16
b. $2,000 / 1.1 = $1,818.18
c. $500 / 1.18 = $233.25
4.3 Would you rather receive $1,000 today or $2,000 in 10 years if the discount rate is 8 percent?
96 Part II Value and Capital Budgeting
14The following conventions are used in the questions and problems for this chapter.
If more frequent compounding than once a year is indicated, the problem will either state: (1) both a stated
annual interest rate and a compounding period, or (2) an effective annual interest rate.
If annual compounding is indicated, the problem will provide an annual interest rate. Since the stated
annual interest rate and the effective annual interest rate are the same here, we use the simpler annual
interest rate.
4.3 You can make your decision by computing either the present value of the $2,000 that you
can receive in ten years, or the future value of the $1,000 that you can receive now.
Present value: $2,000 / 1.0810 = $926.39
Future value: $1,000 1.0810 = $2,158.93
Answers to End-of-Chapter Problems B-51 Either calculation indicates you should take the $1,000 now.
4.4 The government has issued a bond that will pay $1,000 in 25 years. The bond will pay no interim
coupon payments. What is the present value of the bond if the discount rate is 10 percent?
4.4 Since this bond has no interim coupon payments, its present value is simply the present
value of the $1,000 that will be received in 25 years. Note: As will be discussed in the next
chapter, the present value of the payments associated with a bond is the price of that bond.
PV = $1,000 /1.125 = $92.30
4.5 A firm has an estimated pension liability of $1.5 million due 27 years from today. If the
firm can invest in a risk-free security that has a stated annual interest rate of 8 percent, how
much must the firm invest today to be able to make the $1.5 million payment?
4.5 PV = $1,500,000 / 1.0827 = $187,780.23
4.6 You have won the Florida state lottery. Lottery officials offer you the choice of the following
alternative payouts:
Alternative 1: $10,000 one year from now.
Alternative 2: $20,000 five years from now.
Which alternative should you choose if the discount rate is:
a. 0 percent?
b. 10 percent?
c. 20 percent?
d. What discount rate makes the two alternatives equally attractive to you?
4.6 a. At a discount rate of zero, the future value and present value are always the same.
Remember, FV = PV (1 + r) t. If r = 0, then the formula reduces to FV = PV.
Therefore, the values of the options are $10,000 and $20,000, respectively. You
should choose the second option.
b. Option one: $10,000 / 1.1 = $9,090.91
Option two: $20,000 / 1.15 = $12,418.43
Choose the second option.
c. Option one: $10,000 / 1.2 = $8,333.33
Option two: $20,000 / 1.25 = $8,037.55
Choose the first option.
d. You are indifferent at the rate that equates the PVs of the two alternatives. You
know that rate must fall between 10% and 20% because the option you would
choose differs at these rates. Let r be the discount rate that makes you indifferent
between the options.
$10,000 / (1 + r) = $20,000 / (1 + r)5
(1 + r)4 = $20,000 / $10,000 = 2
1 + r = 1.18921
r = 0.18921 = 18.921%
4.7 You are selling your house. The Smiths have offered you $115,000. They will pay you immediately.
The Joneses have offered you $150,000, but they cannot pay you until three years from today. The interest
rate is 10 percent. Which offer should you choose?
4.7 PV of Joneses’ offer = $150,000 / (1.1)3 = $112,697.22
Since the PV of Joneses’ offer is less than Smiths’ offer, $115,000, you should choose
Smiths’ offer.
4.8 Suppose you bought a bond that will pay $1,000 in 20 years. No intermediate coupon
payments will be made. If the appropriate discount rate for the bond is 8 percent,
a. what is the current price of the bond?
b. what will the price be 10 years from today?
c. what will the price be 15 years from today?
4.8 a. P0 = $1,000 / 1.0820 = $214.55
