DCF Solutions Manual Ch06
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1
Chapter 6 Discounted Cash Flows and Valuation Before You Go On Questions and Answers Section 6.1 1.
Explain how to calculate the future value of a stream of cash flows.
It would helpful to first construct a time line so that we can identify the timing of each cash flow. Then you would calculate the future vale of each individual cash flow. Finally, you would add up the future values of all the individual cash flows to determine the future value of the cash flow stream.
2.
Explain how to calculate the present value of a stream of cash flows.
To calculate the present value of a stream of cash flows, you should first draw a time line so that you can see that each cash flow is placed in its correct time period. Then you simply calculate the present value of each cash flow for its time period, and finally you add up all the present values.
3.
Why is it important to adjust all cash flows to a common date? When making economic decisions, we need to compare ―apples to apples.‖ This is possible only when we bring all the cash flows to a common date, which can either be a present time or some future date. The reason is the time value of money: a dollar today is worth more than a dollar in the future. Thus, when cash flows are converted to the same time period, the time value of money concept holds true, and we can concentrate on the economic aspects of the decision.
Section 6.2
2 1.
How do an ordinary annuity, an annuity due, and a perpetuity differ?
Ordinary annuity assumes that the cash flows occur at the end of a period. Most types of loans are ordinary annuities. On the other hand, annuity due is an annuity whose payment is to be made immediately (or at the beginning of a period) instead of at the end of the period. For example, in many leases the first payment is due immediately, and each successive payment must be made at the beginning of the month. Perpetuity is a special case of annuity, and it refers to a constant stream of identical cash flows with no end.
2.
Give two examples of perpetuities. The text gives the example of British government bonds called consols that have no maturity and have been traded in the markets since the end of the Napoleonic wars. Another example could be a preferred stock of a company that has no maturity and will pay a constant dividend forever.
3.
What is the annuity transformation method?
The annuity transformation method refers to the conversion of an ordinary annuity to annuity due. In this process, you first plot all the cash flows on a time line as if the cash flows were an ordinary annuity. Then you calculate the present or future value factor as you would with an ordinary annuity, and finally, you multiple your answer by (1 + i). Conveniently, this relationship works for both present and future value calculations.
Section 6.3 1.
What is the difference between a growing annuity and a growing perpetuity?
A stream of cash flows that is growing at a constant rate over time can be called a growing annuity or growing perpetuity. If the cash flows extend over a finite length of time, then we call it a growing annuity and can use Equation6.5 to compute the present value. If the growth will continue for a very long time period and perhaps, forever, we
3 refer to it as the growing perpetuity. We would then use Equation 6.6 to estimate the present value of this cash flow stream.
Section 6.4 1.
What is the APR, and why are lending institutions required to disclose this rate?
APR, or the annual percentage rate, is the annualized interest rate using simple interest. It is defined as the simple interest charged per period multiplied by the number of compounding periods per year. Lending institutions are mandated by Federal Truth-inLending Act regulations to disclose this rate to essentially make it easier for consumers to be exposed to the same kind of rate by all businesses.
2.
What is the correct way to annualize an interest rate in financial decisions?
The correct way to annualize interest rates is by computing the effective annual interest rate (EAR). This is the annual growth rate that allows for compounding, which means you earn interest on interest. To calculate the EAR, take the quoted rate and divide it by the number of compounding periods (quoted rate/m). Then take the resulting interest rate, add 1 to it, and raise it to the power equal to m. Finally, subtract 1 and the result is EAR.
3.
Distinguish between quoted interest rate, interest rate per period, and effective annual interest rate.
Quoted interest rate, such as APR, is the interest rate that has been annualized by multiplying the rate per period by the number of compounding periods. Interest rate per period is the quoted rate per period. It can be stated in the form of an APR—in that case, just divide it by the number of compounding periods to obtain the interest rate per period. Finally, EAR is the annual rate of interest that accounts for the effects of compounding.
4
Self-Study Problems 6.1
Kronka, Inc., is expecting cash flows of $13,000, $11,500, $12,750, and $9,635 over the next four years. What is the present value of these cash flows if the appropriate discount rate is 8 percent?
Solution: The time line for the cash flows and their present value is as follows: 0
8%
1
2
3
4
Year
├─────────┼─────────┼─────────┼─────────┤ $13,000
$11,500
$12,750
$9,635
$13,000 $11,500 $12,750 $9,635 (1.08) (1.08) 2 (1.08) 3 (1.08) 4 $12,037.03 $9,859.40 $10,121.36 $7,082.01 $39,099.80
PV4
6.2
Your grandfather has agreed to deposit a certain amount of money each year into an account paying 7.25 percent annually to help you go to graduate school. Starting next year, and for the following four years, he plans to deposit $2,250, $8,150, $7,675, $6,125, and $12,345 into the account. How much will you have at the end of the five years?
Solution: The time line for the cash flows and their future value is as follows:
0
7.25%
1
2
3
4
5
├─────────┼─────────┼─────────┼──────────┼─────────┤
-$2,250
-$8,150
-$7,675
-$6,125
-$12,345
Year
5
FV5 [$2, 250 (1.0725) 4 ] [$8,150 (1.0725)3 ] [$7,675 (1.0725)2 ] [$6,125 (1.0725)2 ] $12,345 $2,976.95 $10,054.25 $8,828.22 $6,569.06 $12,345.00 $40, 773.48
6.3
Mike White is planning to save up for a trip to Europe in three years. He will need $7,500 when he is ready to make the trip. He plans to invest the same amount at the end of each of the next three years in an account paying 6 percent. What is the amount he will have to save every year to reach his goal of $7,500 in three years?
Solution: Amount Mike White will need in three years = FVA3 = $7,500 Number of years = n = 3 Interest rate on investment =. i = 6.0% Amount needed to be invested every year = PMT = ?
0
6%
1
2
3
Year
├─────────┼──────┼───────┤ CF=?
CF=?
CF=? FVA3 = $7,500
FVA n CF $7,500 CF
(1 i ) n 1 i (1 0.06)3 1 0.06
CF 3.1836 CF
$7,500 3.1836
$2, 355.82
Mike will have to invest $2,353.82 every year for the next three years.
6
6.4
Becky Scholes has $150,000 to invest. She wants to be able to withdraw $12,500 every year forever without using up any of her principal. What interest rate would her investment have to earn in order for her to be able to so?
Solution: Present value of investment = $150,000 Amount needed annually = $12,500 This is a perpetuity!
PV( Perpetuity)
CF i
CF $12,500 PV( Perpetuity) $150,000 i 8.33%
i
6.5
Dynamo Corp. is expecting annual payments of $34,225 for the next seven years from a customer. What is the present value of this annuity if the discount rate is 8.5 percent?
Solution: 0 8.5% 1
2
3
4
5
6
7
├───┼───┼────┼───┼───┼───┼───┤ PVA= ?
$34,225 $34,225
$34,225 $34,225 $34,225 $34,225 $34,225
PVA 7 CF
1 1 (1 i ) n i
1 1 (1.085) 7 $34, 225 0.085 $34, 225 5.1185 $175,181.14
7
Critical Thinking Questions 6.1
Identify the steps involved in computing the future value when you have multiple cash flows.
First, prepare a time line to identify the size and timing of the cash flows. Second, calculate the present value of each individual cash flow using an appropriate discount rate. Finally, add up the present values of the individual cash flows to obtain the present value of a cash flow stream. This approach is especially useful in the real world where the cash flows for each period are not the same.
6.2
What is the key economic principle involved in calculating the present value and future value of multiple cash flows?
Regardless of whether you are calculating the present value or the future value of a cash flow stream, the key idea is to discount or compound the cash flows to the same point in time.
6.3
What is the difference between a perpetuity and an annuity?
A cash flow stream that consists of the same amount being received or paid on a periodic basis is called an annuity. If the same payments are made periodically forever, the contract is called a perpetuity.
6.4
Define annuity due. Would an investment be worth more if it was an ordinary annuity or an annuity due? Explain.
When annuity cash flows occur at the beginning of each period, it is called an annuity due. Annuity due will result in a bigger investment than an ordinary annuity because each cash flow will accrue an extra interest payment.
8 6.5
Raymond Bartz is trying to choose between two equally risky annuities, each paying $5,000 per year for five years. One is an ordinary annuity, and the other is an annuity due. Which of the following statements is most correct? a. The present value of the ordinary annuity must exceed the present value of the annuity due, but the future value of an ordinary annuity may be less than the future value of the annuity due. b. The present value of the annuity due exceeds the present value of the ordinary annuity, while the future value of the annuity due is less than the future value of the ordinary annuity. c. The present value of the annuity due exceeds the present value of the ordinary annuity, and the future value of the annuity due also exceeds the future value of the ordinary annuity. d. If interest rates increase, the difference between the present value of the ordinary annuity and the present value of the annuity due remains the same.
c. The present value of the annuity due exceeds the present value of the ordinary annuity, and the future value of the annuity due also exceeds the future value of the ordinary annuity.
6.6
Which of the following investments will have the highest future value at the end of three years? Assume that the effective annual rate for all investments is the same. a. You earn $3,000 at the end of three years (a total of one payment). b. You earn $1,000 at the end of every year for the next three years (a total of three payments). c. You earn $1,000 at the beginning of every year for the next three years (a total of three payments).
c. Earning $1,000 at the beginning of each year for the next three years will have the highest future value as it is an annuity due.
6.7
Explain whether or not each of the following statements is correct.
9
a. A 15-year mortgage will have larger monthly payments than a 30-year mortgage of the same amount and same interest rate.
This is a true statement. The 15-year mortgage will have higher monthly payments since more of the principal will have to be paid each month than in the case of a 30-year mortgage.
b. If an investment pays 10 percent interest compounded annually, its effective rate will also be 10 percent.
This is true since the frequency of compounding is annual and hence the rate for a single period is the same as the rate for a year.
6.8
When will the annual percentage rate (APR) be the same as the effective annual rate (EAR)?
The annual percentage rate (APR) will be the same as the effective annual rate only if the compounding period is annual, not otherwise.
6.9
Why is the EAR superior to the APR in measuring the true economic cost or return?
Unlike the APR, which reflects annual compounding, the EAR takes into account the actual number of compounding periods. For example, suppose there are two investment alternatives that both pay an APR of 10 percent. Assume that the first pays interest annually and that the second pays interest quarterly. It would be a mistake to assume that both investments will provide the same return. The real return on the first one is 10 percent, but the second investment actually provides a return of 10.38 percent because of the quarterly compounding. Thus, this is the superior investment!
6.10 Suppose three investments have equal lives and multiple cash flows. A high discount rate
10 tends to favor: a. the investment with large cash flow early. b. the investment with large cash flow late. c. the investment with even cash flow. d. neither investment since they have equal lives.
a. The investment with large cash flows early will be worth more compared to the one with the large cash flows late. The cash flows that come in later will have a heavier penalty when using a higher discount rate. Thus the investment with large cash flows early will be favored.
Questions and Problems
BASIC 6.1
Future value with multiple cash flows: Konerko, Inc., expects to earn cash flows of $13,227, $15,611, $18,970, and $19,114 over the next four years. If the company uses an 8 percent discount rate, what is the future value of these cash flows at the end of year 4? LO 1
Solution: 0
8%
1
2
3
4
├───────┼────────┼───────┼────────┤ $13,227
$15,611
$18,970
$19,114
FV4 [$13, 227 (1.08)3 ] [$15,611 (1.08) 2 ] [$18,970 (1.08)1 ] $19,114 $16,662.21 $18, 208.67 $20, 487.60 $19,114 $74, 472.48
11 6.2
Future value with multiple cash flows: Ben Woolmer has an investment that will pay him the following cash flows over the next five years: $2,350, $2,725, $3,128, $3,366, and $3,695. If his investments typically earn 7.65 percent, what is the future value of the investment’s cash flows at the end of five years? LO 1
Solution: 0
7.65%
1
2
3
4
5
Year
├───────┼────────┼───────┼────────┼───────┤ $2,350
$2,725
$3,128
$3,366
$3,695
FV5 $2,350 (1.0765)4 $2,725 (1.0765)3 $3,128 (1.0765)2 $3,366 (1.0765)1 $3,695 $3,155.91 $3,399.45 $3,624.89 $3,623.50 $3,695 $17, 498.75
6.3
Future value with multiple cash flows: You are a freshman in college and are planning a trip to Europe when you graduate from college at the end of four years. You plan to save the following amounts starting today: $625, $700, $700, and $750. If the account pays 5.75 percent annually, how much will you have at the end of four years? LO 1
Solution: 0
5.75%
1
2
3
4
Year
├───────┼────────┼───────┼────────┤ $625
$700
$700
$750
FV4 $625(1.0575) 4 $700(1.0575) 3 $700(1.0575) 2 $750(1.0575) $781.63 $827.83 $782.81 793.13 $3,185.40
12 6.4
Present value with multiple cash flows: Saul Cervantes has just purchased some equipment for his landscaping business. For this equipment he must pay the following amounts at the end of each of the next five years: $10,450, $8,500, $9,675, $12,500, and $11,635. If the appropriate discount rate is of 10.875 percent, what is the cost of the equipment Saul purchased today? LO 1
Solution: 0
10.875%
1
2
3
4
5
Year
├───────┼────────┼───────┼────────┼───────┤ $10,450
$8,500
$9,675
$12,500
$11,635
$10,450 $8,500 $9,675 $12,500 $11,635 2 3 4 (1.10875) (1.10875) (1.10875) (1.10875) (1.10875) 5 $9,425.03 $6,914.35 $7,098.23 8,271.33 $6,943.82 $38,652.76
PV
6.5
Present value with multiple cash flows: Jeremy Fenloch borrowed some money from his friend and promised to repay him the amounts of $1,225, $1,350, $1,500, $1,600, and $1,600 over the next five years. If the friend normally discounts investments at 8 percent annually, how much did Jeremy borrow? LO 1
Solution: 0
8%
1
2
3
4
5
├───────┼────────┼───────┼────────┼───────┤ $1,225
$1,350
$1,500
$1,600
$1,225 $1,350 $1,500 $1,600 $1,600 (1.08) (1.08) 2 (1.08) 3 (1.08) 4 (1.08) 5 $1,134.26 $1,157.41 $1,190.75 $1,176.05 $1,088.93 $5,747.40
PV
$1,600
Year
13
6.6
Present value with multiple cash flows: Biogenesis, Inc. management expects the following cash flow stream over the next five years. The company discounts all cash flows using a 23 percent discount rate. What is the present value of this cash flow stream? LO 1
1
2
3
4
5
Year
┼────────┼───────┼────────┼───────┤ -$1,133,676
-$978,452
$275,455
$878,326
$1,835,444
1
2
3
4
Solution: 0
23%
5
Year
├───────┼────────┼───────┼────────┼───────┤ -$1,133,676
-$978,452
$275,455
$878,326
$1,835,444
$1,133,676 $978,452 $275,455 $878,326 $1,835,444 (1.23) (1.23) 2 (1.23) 3 (1.23) 4 (1.23) 5 $921,687.80 $646,739.37 $148,025.09 $383,738.43. $651,951.94 $384,711.72
PV
6.7
Present value of an ordinary annuity: An investment opportunity requires a payment of $750 for 12 years, starting a year from today. If your required rate of return is 8 percent, what is the value of the investment today? LO 2
Solution: 0
8%
1
2
3
11
12
├───────┼────────┼───────┼………………┼───────┤ $750
$750
$750
$750
$750
Year
14
Annual payment = PMT = $750 No. of payments = n = 12 Required rate of return = 8% Present value of investment = PVA12 1
1 (1 i ) n i
1
1 (1.08)12 $750 7.5361 0.08
PVA n PMT
$750 $5, 652.06
6.8
Present value of an ordinary annuity: Dynamics Telecommunications Corp. has made an investment in another company that will guarantee it a cash flow of $22,500 each year for the next five years. If the company uses a discount rate of 15 percent on its investments, what is the present value of this investment? LO 2
Solution: 0
15%
1
2
3
4
5
├───────┼────────┼───────┼────────┼───────┤ $22,500
$22,500
Annual payment = PMT = $22,500 No. of payments = n = 5 Required rate of return = 15% Present value of investment = PVA5
$22,500
$22,500
$22,500
15 1 PVA n PMT
1 (1 i ) n i 1
$22,500
1 (1.15)5 $22,500 3.3522 0.15
$75, 423.49
6.9
Future value of an ordinary annuity: Robert Hobbes plans to invest $25,000 a year at the end of each year for the next seven years in an investment that will pay him a rate of return of 11.4 percent. How much money will Robert will have at the end of seven years? LO 2
Solution: 0
11.4%
1
2
3
6
7
Year
├───────┼────────┼───────┼………………┼───────┤ $25,000
$25,000
$25,000
$25,000
$25,000
Annual investment = PMT = $25,000 No. of payments = n = 7 Investment rate of return = 11.4% Future value of investment = FVA7
FVA n PMT
(1 i ) n 1 i
(1.114)7 1 $25,000 $25,000 9.9044 0.114 $247, 609.95
6.10
Future value of an ordinary annuity: Cecelia Thomas is a sales executive at a Baltimore firm. She is 25 years old and plans to invest $3,000 every year in an IRA account, beginning at the end of this year until she turns 65 years old. If the IRA
16 investment will earn 9.75 percent annually, how much will she have in 40 years when she turns 65 years old? LO 2
Solution: 0
9.75%
1
2
3
39
40
Year
├───────┼────────┼───────┼………………┼───────┤ $3,000
$3,000
$3,000
$3,000
$3,000
Annual investment = PMT = $3,000 No. of payments = n = 40 Investment rate of return = 9.75% Future value of investment = FVA40
FVA n PMT
(1 i ) n 1 i
$3,000
(1.0975)40 1 $3,000 413.5588 0.0975
$1, 240, 676.41
6.11
Future value of an annuity. Refer to Problem 6.10. If Cecelia Thomas starts saving at the beginning of each year, how much will she have at age 65? LO 2
Solution: 0
9.75%
1
2
3
39
40
├───────┼────────┼───────┼………………┼───────┤ $3,000
$3,000
$3,000
Annual investment = PMT = $3,000 No. of payments = n = 40 Type of annuity = Annuity due
$3,000
$3,000
Year
17 Investment rate of return = 9.75% Future value of investment = FVA40
FVA n PMT
(1 i ) n 1 (1 i ) i
$3,000
(1.0975)40 1 (1.0975) $3,000 413.5588 1.0975 0.0975
$1, 361, 642.36
6.12
Computing annuity payment: Kevin Winthrop is saving for an Australian vacation in three years. He estimates that he will need $5,000 to cover his airfare and all other expenses for a week-long holiday in Australia. If he can invest his money in an S&P 500 equity index fund that is expected to earn an average return of 10.3 percent over the next three years, how much will he have to save every year, starting at the end of this year? LO 2
Solution: 0
10.3%
1
2
3
├───────┼────────┼───────┤ PMT
PMT
PMT
FVAn = $5,000 Future value of annuity = FVA = $5,000 Return on investment = i = 10.3% Payment required to meet target = PMT Using the FVA equation:
Year
18
FVA n PMT
(1 i ) n 1 i
$5,000 PMT
(1.103)3 1 0.103
PMT
$5,000 $5,000 3 (1.103) 1 3.3196 0.103
$1, 506.20
Kevin has to save $1,506.20 every year for the next three years to reach his target of $5,000.
6.13
Computing annuity payment: The Elkridge Bar & Grill has a seven-year loan of $23,500 with Bank of America. It plans to repay the loan in seven equal installments starting today. If the rate of interest is 8.4 percent, how much will each payment be? LO 2
0
1
2
3
6
7
├───────┼────────┼───────┼………………┼───────┤ PMT PVAn = $23,500
PMT
PMT n = 7;
Present value of annuity = PVA = $23,500 Return on investment = i = 8.4% Payment required to meet target = PMT Type of annuity = Annuity due Using the PVA equation:
PMT i = 8.4%
PMT
PMT
Year
19
1 PVA n PMT PMT
1 (1 i ) n (1 i ) i
$23,500 $23,500 1 5.1359 1.084 1 7 (1.084) (1.084) 0.084
$4, 221.07 Each payment made by Elkridge Bar & Grill will be $4,221.07, starting today.
6.14
Perpetuity: Your grandfather is retiring at the end of next year. He would like to ensure that his heirs receive payments of $10,000 a year forever, starting when he retires. If he can earn 6.5 percent annually, how much does your grandfather need to invest to produce the desired cash flow? LO 3
Solution: Annual payment needed = PMT = $10,000 Investment rate of return = i = 6.5% Term of payment = Perpetuity Present value of investment needed = PV PMT $10,000 i 0.065 $153,846.15
PV of Perpetuity
6.15
Perpetuity: Calculate the annual cash flows for each of the following investments:
a.
$250,000 invested at 6%
b.
$50,000 invested at 12%
c.
$100,000 invested at 10%
LO 3
20 Solution: a.
Annual payment = PMT Investment rate of return = i = 6% Term of payment = Perpetuity Present value of investment needed = PV = $250,000
PM T i PM T PV i $250,000 0.06 $15,000
PV of Perpetuity
b.
Annual payment = PMT Investment rate of return = i = 12% Term of payment = Perpetuity Present value of investment needed = PV = $50,000
PM T i PM T PV i $50,000 0.12 $6,000
PV of Perpetuity
c.
Annual payment = PMT Investment rate of return = i = 10% Term of payment = Perpetuity Present value of investment needed = PV = $100,000
PM T i PM T PV i $100,000 0.10 $10,000
PV of Perpetuity
6.16. Effective annual interest rate: Raj Krishnan bought a Honda Civic for a price of $17,345. He put down $6,000 and financed the rest through the dealer at an APR of 4.9 percent for four years. What is the effective annual rate (EAR) if payments are made monthly? LO 5
21
Solution: Loan amount = PV = $11,345 Interest rate on loan = i = 4.9% Frequency of compounding = m = 12 Effective annual rate = EAR m1
12
i 0.049 EAR 1 1 1 1 12 m 1.05 1 5%
6.17
Effective annual interest rate: Cyclone Rentals borrowed $15,550 from a bank for three years. If the quoted rate (APR) is 6.75 percent, and the compounding is daily, what is the effective annual rate (EAR)? LO 5
Solution: Loan amount = PV = $15,550 Interest rate on loan = i = 6.75% Frequency of compounding = m = 365 Effective annual rate = EAR m1
i 0.0675 EAR 1 1 1 365 m 1.0698 1 7%
6.18
365
1
Growing perpetuity: You are evaluating a growing perpetuity product from a large financial services firm. The investment promises an initial payment of $20,000 at the end of this year and subsequent payments which will grow at a rate of 3.4 percent annually. If you use a 9 percent discount rate for investment like this, what is the present value of this growing perpetuity? LO 4
Solution:
22 Cash flow at t = 1 = CF1 = $20,000 Annual growth rate = g = 3.4% Discount rate = i = 9% Present value of growing perpetuity = PVA∞
CF1 $20,000 (i g) (0.09 0.034) $357,142.86
PVA
INTERMEDIATE
6.19
Future value with multiple cash flows: Trigen Corp. management will invest cash flows of $331,000, $616,450, $212,775, $818,400, $1,239,644, and $1,617,848 in research and development over the next six years. If the appropriate interest rate is 6.75 percent, what is the future value of these investment cash flows six years from today? LO 1
Solution: 0
6.75%
1
2
3
4
5
6
├───────┼────────┼───────┼────────┼───────┼────────┤ $331,000
$616,450
$212,775
$818,400
$1,239,644
$1,617,848
FV6 $331,000 (1.0675)5 $616, 450 (1.0675)4 $212,775 (1.0675)3 $818, 400 (1.0675)2 $1, 239,644 (1.0765)1 $1,617,848 $458,846.49 $800,514.85 $258,835.74 $932,612.84 $1,323,319.97 $1,617,848 $5, 391, 977.89
6.20
Future value with multiple cash flows: Stephanie Watson plans to adopt the following investment pattern beginning next year. She will invest $3,125 in each of the next three
23 years and will then make investments of $3,650, $3,725, $3,875, and $4,000 over the following four years. If the investments are expected to earn 11.5 percent annually, how much will she have at the end of the seven years? LO 1
Solution: Expected rate of return = i = 11.5% Investment period = n = 7 years Future value of investment = FV
FV7 $3,125 (1.115)6 $3,125 (1.115)5 $616, 450 (1.115)4 $3,650 (1.115)3 $3,725 (1.115)2 $3,875 (1.115)1 $4,000 $6,004.81 $5,385.48 $4,830.03 $5,059.61 $4,631.01 $4,320.63 $4,000 $34, 231.57 6.21
Present value with multiple cash flows: Carol Jenkins, a lottery winner, will receive the following payments over the next seven years. If she can invest her cash flows in a fund that will earn 10.5 percent annually, what is the present value of her winnings? LO 1
0
1
2
3
4
5
6
7
Year
├────┼─────┼─────┼──────┼─────┼──────┼─────┤ $200,000
$250,000 $212,775 $275,000
Solution: Expected rate of return = i = 10.5% Investment period = n = 7 years Future value of investment = FV
$300,000
$350,000
$400,000 $550,000
24
$200,000 $250,000 $275,000 $300,000 $350,000 $400,000 $550,000 (1.105)1 (1.105) 2 (1.105) 3 (1.105) 4 (1.105) 5 (1.105) 6 (1.105) 7 $180,995.48 $204,746.01 $203,819.56 $201,220.46 $212,449.96 $219,728.47 $273,417.77 $1,496,377.71
FV7
6.22
Computing annuity payment: Gary Whitmore is a high school sophomore. He currently has $7,500 in a savings account that pays 5.65 percent annually. Gary plans to use his current savings plus what he can save over the next four years to buy a car. He estimates that the car will cost $12,000 in four years. How much money should Gary save each year if he wants to buy the car? LO 2
Solution: Cost of car in four years = $12,000 Amount invested in money market account now = PV = $7,500 Return earned by investment = i = 5.65% Value of current investment in 4 years = FV4
FV4 PV(1 i ) 4 $7,500(1.0565) 4 $9,344.14 Balance of money needed to buy car = $12,000 – $9,344.14 = $2,655.86 = FVA Payment needed to reach target = PMT
FVA PMT PMT
(1 i ) n 1 i
FVA $2,655.86 $2,655.86 n 4 1 (1 i ) (1.0565) 1 4.351949 i 0.0565
$610.27
6.23
Growing annuity: Modern Energy Company owns several gas stations. Management is looking to open a new station in the western suburbs of Baltimore. One possibility they are evaluating is to take over a station located at a site that has been leased from the
25 county. The lease, originally for 99 years, currently has 73 years before expiration. The gas station generated a net cash flow of $92,500 last year, and the current owners expect an annual growth rate of 6.3 percent. If Modern Energy uses a discount rate of 14.5 percent to evaluate such businesses, what is the present value of this growing annuity? LO 4
Solution: Time for lease to expire = n = 73 years Last year’s net cash flow = CF0 = $92,500 Expected annual growth rate = g = 6.3% Firm’s required rate of return = i = 14.5% Expected cash flow next year = CF1 = $92,500(1 + g) = $92,500(1.063) = $98,327.50 Present value of growing annuity = PVAn
1 g n 1.063 73 CF1 $98,327.50 PVA n 1 1 (i g) 1 i (0.145 0.063) 1.145 $1,199,115.85 0.995593 $1,193,831.54
6.24
Future value of an annuity due: Jeremy Denham plans to save $5,000 every year for the next eight years, starting today. At the end of eight years, Jeremy will turn 30 years old and plans to use his savings toward the down payment on a house. If his investment in a mutual fund will earn him 10.3 percent annually, how much will he have saved in eight years when he buys his house? LO 2
Solution: 0
10.3%
1
2
3
7
8
├───────┼────────┼───────┼………………┼───────┤ $5,000
$5,000
$5,000
$5,000
$5,000
Year
26 Annual investment = PMT = $5,000 No. of payments = n = 8 Type of annuity = Annuity due Investment rate of return = 10.3% Future value of investment = FVA8
(1 i ) n 1 FVA n PM T (1 i ) i (1.103) 8 1 $5,000 (1.103) $5,000 11.5612 1.103 0 . 103 $63,760.19
6.25
Present value of an annuity due: Grant Productions has borrowed a large sum from the California Finance Company at a rate of 17.5 percent for a seven-year period. The loan calls for a payment of $1,540,862.19 each year beginning today. How much did Grant borrow? LO 2
Solution: 0
17.5%
1
2
3
6
7
├───────┼────────┼───────┼………………┼───────┤ PMT =$1,540,862.19 at the beginning of each year
Annual payment = PMT = $1,540,862.19 Type of annuity = Annuity due No. of payments = n = 7 Required rate of return = 17.5% Present value of investment = PVA8
27
1 PVA n PMT
1 (1 i ) n (1 i ) i 1
$1,540,862.19
1 (1.175)7 (1.175) $1,540,862.19 3.8663 1.175 0.175
$6, 999, 999.98 $7, 000, 000
6.26
Present value of an annuity due: Sharon Kabana has won a state lottery and will receive a payment of $89,729.45 every year, starting today, for the next 20 years. If she invests the proceeds at a rate of 7.25 percent, what is the present value of the cash flows that she will receive? Round to the nearest dollar. LO 2
Solution: 0
7.25%
1
2
3
19
20
├───────┼────────┼───────┼………………┼───────┤ PMT = $89,729.45
at the beginning of each year
Annual payment = PMT = $89,729.45 Type of annuity = Annuity due No. of payments = n = 20 Required rate of return = 7.25% Present value of investment = PVA20
1 PVA n PMT
1 (1 i ) n (1 i ) i 1
$89,729.45
1 (1.0725) 20 (1.0725) $89,729.45 10.3912 1.0725 0.0725
$999, 999.95 $1, 000, 000
28 6.27
Present value of an annuity due: You wrote a piece of software that does a better job of allowing computers to network than any other program designed for this purpose. A large networking company wants to incorporate your software into their systems and is offering to pay you $500,000 today, plus $500,000 at the end of each of the following six years for permission to do this. If the appropriate interest rate is 6 percent, what is the present value of the cash flow stream that the company is offering you? LO 2
Solution:
You are being offered a seven year annuity due. You can solve this problem several different ways. First, you can calculate the present value of each of the individual cash flows and then add the present values together. Second, you can use the annuity transformation method discussed in the chapter. Third, you can calculate the present value of an ordinary six-year annuity and then add $500,000 to that value. Of course you can also use your calculator or Excel to do the calculations for you. Below are the calculations for the annuity transformation method.
PVA7 = CF × PV annuity factor = $500,000 × 5.582 = $2,791,000
Annuity value due
= PVA7 × (1 + i) = $2,791,000 × 1.06 = $2,958,460
6.28
Present value of an annuity: Suppose that the networking company in problem 6.27 will not start paying you until the first of new systems that use your software is sold in two years. What is the present value of that annuity? Assume that the appropriate interest rate is still 6 percent. LO 2
29
Solution:
Since the first cash flow will not be received for two years, the present value will be less than in problem 6.27. All you have to do to solve this problem is to discount the answer from problem 6.27 for two years. Annuity value = value of annuity value due/(1 + i)2 = $2,958,460/(1.06)2 = $2,633,019
This is also equal to the value of the regular annuity in problem 6.27, PVA7, discounted one year at 6 percent.
6.29
Perpetuity: Calculate the present value of the following perpetuities: a. $1,250 discounted back to the present at 7% b. $7,250 discounted back to the present at 6.33% c. $850 discounted back to the present at 20% LO 3
Solution: a.
Annual payment = PMT =$1,250 Investment rate of return = i = 7% Term of payment = Perpetuity Present value of investment needed = PV PMT $1,250 i 0.07 $17,857.14
PV of Perpetuity
b.
Annual payment = PMT =$7,250 Investment rate of return = i = 6.33% Term of payment = Perpetuity.
30 Present value of perpetuity = PV PMT $7,250 i 0.0633 $114,533.97
PV of Perpetuity
c.
Annual payment = PMT =$850 Investment rate of return = i = 20% Term of payment = Perpetuity. Present value of investment needed = PV PMT $850 i 0.20 $4,250
PV of Perpetuity
6.30
Effective annual interest rate: Find the effective annual interest rate (EAR) on each of the following: a. 6% compounded quarterly. b. 4.99% compounded monthly. c. 7.25% compounded semi-annually. d. 5.6% compounded daily. LO 5
Solution: a.
Interest rate = i = 6% Frequency of compounding = m = 4 Effective annual rate = EAR m1
4
i 0.06 EAR 1 1 1 1 4 m 1.06136 1 6.14%
b.
Interest rate = i = 4.99% Frequency of compounding = m = 12 Effective annual rate = EAR
31 m1
12
i 0.0499 EAR 1 1 1 1 12 m 1.0511 1 5.11%
c.
Interest rate = i = 7.25% Frequency of compounding = m = 2 Effective annual rate = EAR m1
2
i 0.0725 EAR 1 1 1 1 2 m 1.0738 1 7.38%
d.
Interest rate = i = 5.6% Frequency of compounding = m = 365 Effective annual rate = EAR m1
i 0.056 EAR 1 1 1 365 m 1.0576 1 5.76%
6.31
365
1
Effective annual interest rate: Which of the following investments has the highest effective annual rate (EAR)?
a. A bank CD that pays 8.25% interest quarterly. b. A bank CD that pays 8.25% monthly. c. A bank CD that pays 8.45% annually. d. A bank CD that pays 8.25% semiannually. e. A bank CD that pays 8% daily (on a 365-day basis). LO 5
Solution: a.
Interest rate on CD = i = 8.25% Frequency of compounding = m = 4 Effective annual rate = EAR
32 m1
4
i 0.0825 EAR 1 1 1 1 4 m 1.08509 1 8.51%
b.
Interest rate on CD = i = 8.25% Frequency of compounding = m = 12 Effective annual rate = EAR m1
12
i 0.0825 EAR 1 1 1 1 12 m 1.0857 1 8.57%
c.
Interest rate on CD = i = 8.45% Frequency of compounding = m = 12 Effective annual rate = EAR m1
1
i 0.0845 EAR 1 1 1 1 1 m 1.0845 1 8.45%
d.
Interest rate on CD = i = 8.25% Frequency of compounding = m = 2 Effective annual rate = EAR m1
2
i 0.0825 EAR 1 1 1 1 2 m 1.0842 1 8.42%
e.
Interest rate on CD = i = 8% Frequency of compounding = m = 365 Effective annual rate = EAR m1
i 0.08 EAR 1 1 1 m 365 1.0833 1 8.33%
365
1
The bank CD that pays 8.25 percent monthly has the highest yield.
33
6.32
Effective annual interest rate: You are considering three alternative investments: (1) a three-year bank CD paying 7.5 percent interest compounded quarterly; (2) a three-year bank CD paying 7.3 percent interest compounded monthly; and (3) a three-year bank CD paying 7.75 percent interest compounded annually. Which investment has the highest effective annual rate? LO 5
Solution: (1)
Interest rate on CD = i = 7.5% Frequency of compounding = m = 4 Effective annual rate = EAR m1
4
i 0.075 EAR 1 1 1 1 4 m 1.0771 1 7.71%
(2)
Interest rate on CD = i = 7.3% Frequency of compounding = m = 12 Effective annual rate = EAR m1
12
i 0.073 EAR 1 1 1 1 12 m 1.0755 1 7.55%
(3)
Interest rate on CD = i = 7.75% Frequency of compounding = m = 1 Effective annual rate = EAR m1
1
i 0.0775 EAR 1 1 1 1 1 m 1.0775 1 7.75%
34 The three-year bank CD paying 7.75 percent interest compounded annually has the highest effective yield.
ADVANCED 6.33
You have been offered the opportunity to invest in a project which is expected to provide you with the following cash flows: $4,000 in 1 year, $12,000 in 2 years, and $8,000 in 3 years. If the appropriate interest rates are 6 percent for the first year, 8 percent for the second year, and 12 percent for the third year, what is the present value of these cash flows?
LO 1 Solution: The answer is $20,495.15:
PV
$4,000 $12,000 $8,000 1.06 1.061.08 1.06 1.081.12
$3,773.58 $10, 482.18 $6, 239.39 $20, 495.15 If the interest rate is different each year, then you must perform the calculation in this way. What you are effectively doing here is discounting each cash flow one year at a time using the appropriate rate. For example, you discount the year three cash flow of $8,000 for one year at 12 percent:
$8,000/1.12 = $7,142.86
You then discount this result for one year at 8 percent:
$7,142.86/1.08 = $6,613.76
35 Finally, you get the present value of the $8,000 by discounting the above result by 6 percent:
$6,613.76/1.06 = $6,239.40
The one cent difference from the number above is due to rounding.
6.34
Tirade Owens, a professional athlete, currently has a contract that will pay him a large amount in the first year of his contract and smaller amounts thereafter. He and his agent have asked the team to restructure the contract. The team, though reluctant, obliged. Tirade and his agent came up with a counteroffer. What are the present values of each of the contracts using a 14 percent discount rate? Which of the three contacts has the highest present value?
Year
Current Contract
Team’s Offer
1
$8,125,000
$4,000,000.00
$5,250,000.00
2
$3,650,000
$3,825,000.00
$7,550,000.00
3
$2,715,000
$3,850,000.00
$3,625,000.00
4
$1,822,250
$3,925,000.00
$2,800,000.00
Counteroffer
LO 1
Solution:
Current Contract
$8,125,000 $3,650,000 $2,715,000 $1,822,250 (1.14) (1.14) 2 (1.14) 3 (1.14) 4 $7,127,192.98 $2,808,556.48 $1,832,547.67 $1,078,918.29 $12,847,215.41
PV
36 Team’s Offer
$4,000,000 $3,825,000 $3,850,000 $3,925,000 (1.14) (1.14) 2 (1.14) 3 (1.14) 4 $3,508,771.93 $2,943,213.30 $2,598,640.34 $2,323,915.09 $11,374,540.65
PV
Counteroffer
$5,250,000 $7,550,000 $3,625,000 $2,800,000 (1.14) (1.14) 2 (1.14) 3 (1.14) 4 $4,605,263.16 $5,809,479.84 $2,446,771.75 $1,657,824.78 $14,519,339.52
PV
The counteroffer has the best value for the player.
6.35
Gary Kornig will turn 30 years old next year and wants to retire when his is 65. So far he has saved (1) $6,950 in an IRA account in which his money is earning 8.3 percent annually and (2) $5,000 in a money market account in which he is earning 5.25 percent annually. Gary wants to have $1 million when he retires. Starting next year, he plans to invest a fixed amount every year until he retires in a mutual fund in which he expects to earn 9 percent annually. How much will Gary have to invest every year to achieve his savings goal? LO 1, LO 2
Solution: Investment (1) Balance in IRA investment = PV = $6,950 Return on IRA account = i = 8.3% Time to retirement = n = 35 years Value of IRA at age 65 = FVIRA FVIRA PV (1 i ) n $6,950 (1.083) 35 $113, 235.03
37
Investment (2) Balance in money market investment = PV = $5,000 Return on money market account = i = 5.25% Time to retirement = n = 35 years Value of money market at age 65 = FVMMA FVMMA PV (1 i ) n $5,000 (1.0525) 35 $29,973.93
Target retirement balance = $1,000,000 Future value of current savings = $113,235.03 + $29,973.93 = $143,208.96 Amount needed to reach retirement target = FVA = $856,774.04 Annual payment needed to meet target = PMT Expected return from mutual fund = i = 9%
(1 i ) n 1 FVA PMT i PMT
FVA $856,791.04 $856,791.04 n 215.711 (1 i ) 1 (1.09)35 1 i 0.09
$3, 971.94
6.36
The top prize for the state lottery is $100,000,000. You have decided it is time for you to take a chance and purchase a ticket. Before you purchase the ticket, you must decide whether to choose the cash option or the annual payment option. If you choose the annual payment option and win, you will receive $100,000,000 in 25 equal payments of $4,000,000─one payment today and one payment at the end of each of the next 24 years. If you choose the cash payment, you will receive a one-time lump sum payment of $59,194,567.18. If you can invest the proceeds and earn 6 percent, which option should you choose? LO 2
38 Solution: You should choose the lump sum option. To see this, all you have to do is calculate the present value of the cash flows from the annual payment option using the 6 percent discount rate. Since the annual payment option is like a 25- year annuity due, this value is:
PVA25 = CF × PV annuity factor = $4,000,000 × 12.783 = $51,132,000
Annuity value due
= PVA25 × (1 + i) = $51,132,000 × 1.06 = $54,199,920
Since this is less than the $59,194,567.18 lump sum, you are better off taking the lump sum.
6.37
At what interest rate would you be indifferent between the cash and annual payment options in problem 6.36?
LO 2
Solution:
You would be indifferent if you could only earn 5 percent from your investments. With this discount rate, the present value of the cash flows from the annual payment option equals the $59,194,567.18 lump sum.
PVA25 = CF × PV annuity factor = $4,000,000 × 14.094 = $56,376,000
Annuity value due
= PVA25 × (1 + i) = $56,376,000 × 1.06
39 = $59,758,560
The difference between the $59,194,567.18 lump sum and $59,758,560 is due to rounding. If you use your calculator or Excel to calculate the present value of the cash flows from the annual payment option you will get exactly $59,194,567.18.
6.38
Babu Baradwaj is saving for his son’s college tuition. His son is currently 11 years old and will begin college in seven years. He has an index fund investment of $7,500 earning 9.5 percent annually. Total expenses at the University of Maryland, where his son says he plans to go, currently total $15,000 per year but are expected to grow at roughly 6 percent each year. Babu plans to invest in a mutual fund that will earn 11 percent annually to make up the difference between the college expenses and his current savings. In total, Babu will make seven equal investments with the first starting today and with the last being made a year before his son begins college. a. What will be the present value of the four years of college expenses at the time that Babu’s son starts college? Assume a discount rate of 5.5 percent. b. What will be the value of the index mutual fund when his son just starts college? c. What is the amount that Babu will have to have saved when his son turns 18 if Babu plans to cover all of his son’s college expenses? d. How much will Babu have to invest every year in order for him to have enough funds to cover all his son’s expenses? LO 1, LO 2
Solution: Annual cost of college tuition today (t = 0) = $15,000 Expected increase in annual tuition costs = g = 6% a.
Four-year tuition costs (t = 7 to t = 10)
Years from now
Future value calculation
Tuition costs
7
$15,000(1.06)7
$22,554.45
40 8
$15,000(1.06)8
$23,907.72
9
$15,000(1.06)9
$25,342.18
10
$15,000(1.06)10
$26,862.72
Discount rate = i = 5.5% Present value of tuition costs = PV $22,554.45 $23,907.72 $25,342.18 $26,862.72 (1.055) (1.055) 2 (1.055) 3 (1.055) 4 $21,378.63 $21,479.95 $21,581.75 $21,684.03 $86,124.36
PV
b.
Future value of the index mutual fund at t = 7 Present value of index fund investment = PV = $7,500 Return on fund = i = 9.5% Future value of investment = FV FV PV (1 i ) n $7,500 (1.095)7 $14,156.64
c.
Target savings needed at t = 7 PV of tuition costs – Future value of investment
= $86,124.36 – $14,156.64 = $71,967.72
d.
Annual savings needed Return on fund = i = 11% Amount that needs to be saved = FVA = $71,967.72 Annuity payment needed = PMT
41
(1 i ) n 1 FVA PMT (1 i ) i PMT
FVA $71,967.72 $71,967.72 7 (1 i ) 1 (1.11) 1 9.7833 1.11 (1 i ) (1.11) i 0.11 n
$6, 627.21
6.39
You are now 50 years old and plan to retire at age 65. You currently have a stock portfolio worth $150,000, a 401(k) retirement plan worth $250,000, and a money market account worth $50,000. Your stock portfolio is expected to provide you annual returns of 12 percent, your 401(k) investment will earn you 9.5 percent annually, and the money market account earns 5.25 percent, compounded monthly. a. If you do not save another penny, what will be the total value of your investments when you retire at age 65? b. Assume you plan to invest $12,000 every year in your 401(k) plan for the next 15 years (starting one year from now). How much will your investments be worth when you retire at 65? c. Assume that you expect to live 25 years after you retire (until age 90). Today, at age 50, you now take all of your investments and place them in an account that pays 8 percent (use the scenario from part b in which you continue saving). If you start withdrawing funds starting at age 66, how much can you withdraw every year (e.g., an ordinary annuity) and leave nothing in your account after a 25th and final withdrawal at age 90? d. You want your current investments, which are described in the problem statement, to support a perpetuity that starts a year from now. How much can you withdraw each year without touching your principal? LO 1, LO 2, LO 3
Solution: a.
Stock Portfolio Current value of stock portfolio = $150,000 Expected return on portfolio = i = 12%
42 Time to retirement = n = 15 years Expected value of portfolio at age 65 = FVStock
FVStock PV (1 i )15 $150,000 (1.12)15 $821,034.86
410(k) Investment Current value of 410(k) portfolio = $250,000 Expected return on portfolio = i = 9.5% Time to retirement = n = 15 years Expected value of portfolio at age 65 = FV401k
FV401k PV (1 i )15 $250,000 (1.095)15 $975,330.48
Money market account Current value of savings = $50,000 Expected return on portfolio = i = 5.25% Time to retirement = n = 15 years Frequency of compounding = m = 12 Expected value of portfolio at age 65 = FVMMA
FVMMA
i PV 1 m
m n
1215
0.0525 $50,000 1 12
$50,000 2.1941 $109,706.14
Total value of all three investments = $821,034.86 + $975,330.48 + $109,706.14 = $1,906,071.48
b.
Planned annual investment in 401k plan = $12,000 Future value of annuity = FVA
(1 i ) n 1 FVA n PM T i (1.095)15 1 $12,000 $12,000 30.5402 0.095 $366,482.77
43 Total investment amount at retirement = $1,906,071.48 + $366,482.77 = $2,272,554.25
c.
Amount available at retirement = PVA = $2,272,554.25 Length of annuity = n = 25 Expected return on investment = i = 8% Annuity amount expected = PMT Using the PVA equation: 1 1 (1 i ) n PVA n PM T i $2,272,554.25 $2,272,554.25 PM T 1 10.6748 1 (1.08) 25 0.08 $212,889.63
Each payment received for the next 25 years will be $212,889.63.
d.
Type of payment = Perpetuity Present value of perpetuity = PVA = $2,272,554.25 Expected return on investment = i = 8%
PM T i PM T $2,272,553.75 0.08 PM T $2,272,554.25 0.08 $181,804.34
PV of Perpetuity
You could receive an annual payment of $181,804.34 forever.
6.40
Trevor Diaz is looking to purchase a Mercedes Benz SL600 Roadster, which has an invoice price of $121,737 and a total cost of $129,482. Trevor plans to put down $20,000
44 and will pay the rest by taking on a 5.75 percent five-year bank loan. What is the monthly payment on this auto loan? Prepare an amortization table using Excel. LO 2
Solution: Cost of new car = $129,482 Down payment = $20,000 Loan amount = $129,482 – $20,000 = $109,482 Interest rate on loan = i = 5.75% Term of loan = n = 5 years Frequency of payment = m = 12 Monthly payment on loan = PMT
1 PVA n PMT PMT
1 (1 i ) n i
$109, 482 $109, 482 1 52.0379 1 125 0575 1 12 0.0575 12
$2,103.89
6.41
The Sundarams are buying a new 3,500-square-foot house in Muncie, Indiana, and will borrow $237,000 from Bank One at a rate of 6.375 percent for 15 years. What is their monthly loan payment? Prepare an amortization schedule using Excel. LO 2
Solution: Home loan amount = $237,000 Interest rate on loan = i = 6.375% Term of loan = n = 15 years
45 Frequency of payment = m = 12 Monthly payment on loan = PMT
1 PVA n PMT PMT
6.42
1 (1 i ) n i
$237,000 1 1 1215 0.06375 1 12 0.06375 12 $237,000 $2, 048.27 115.7072
Assume you will start working as soon as you graduate from college. You plan to start saving for your retirement on your 25th birthday and retire on your 65th birthday.. After retirement, you expect to live at least until you are 85. You wish to be able to withdraw $40,000 (in today’s dollars) every year from the time of your retirement until you are 85 years old (i.e., for 20 years). The average inflation rate is likely to be 5 percent. a. Calculate the lump sum you need to have accumulated at age 65 to be able to draw the desired income. Assume that your return on the portfolio investment is likely to be 10 percent. b. What is the dollar amount you need to invest every year, starting at age 26 and ending at age 65 (i.e., for 40 years) to reach the target lump sum at age 65? c. Now answer questions a and b assuming your rate of return to be 8 percent per year, and then 15 percent per year. d. Now assume you start investing for your retirement when you turn 30 years old and analyze the situation under rate of return assumptions of (i) 8 percent, (ii) 10 percent, and (iii) 15 percent. e. Repeat the analysis by assuming that you start investing only when you are 35 years old. LO 1, LO 2
46
47 Solution:
RETIREMENT ANALYSIS SUMMARY
INVESTMENT AGE = 25
Rate of Return
8%
10%
15%
INVESTMENT AGE = 30
8%
Inflation rate
10%
15%
INVESTMENT AGE = 35
8%
10%
15%
5%
Retirement
$40,000
Income Level
Lump sum needed at age
$5,160,266
$4,353,087
$3,011,353
$5,160,266
$4,353,087
$3,011,353
$5,160,266
$4,353,087 $3,011,353
$19,919
$9,835
$1,693
$29,946
$16,062
$3,417
$45,552
$26,463
65 Annuity payment needed
$6,927
48
Sample Test Problems 6.1
Groves Corp. is expecting annual cash flows of $225,000, $278,000, $312,500, and $410,000 over the next four years. If it uses a discount rate of 6.25 percent, what is the present value of this cash flow stream?
Solution: 0
6.25%
1
2
3
4
├───────┼────────┼───────┼────────┤ $225,000
$278,000
$312,500
$410,000
$225,000 $278,000 $312,500 $410,000 (1.0625) (1.0625) 2 (1.0625) 3 (1.0625) 4 $211,764.71 $246,256.06 $260,533.28 $321,712.62 $1,040,266.67
PV
6.2
Freisinger, Inc., is expecting a new project to start paying off, beginning at the end of next year. It expects cash flows to be as follows:
1
2
$433,676
3
$478,452
4
$475,455
5
$478,326 $535,444
If Freisinger can reinvest these cash flows to earn a return of 7.8 percent, what is the future value of this cash flow stream at the end of five years?
Solution: 0
7.8%
1
2
3
4
5
├───────┼────────┼───────┼────────┼───────┤ $433,676
$478,452
$475,455
$478,326
$535,444
49
FV5 $433,676 (1.078)4 $478, 452 (1.078)3 $475, 455 (1.078)2 $478,326 (1.078)1 $535, 444 $585,653.08 $599,369.52 $552,518.65 $515,635.43 $535, 444 $2, 788, 620.68
6.3
Sochi, Russia, is the site of the next Winter Olympics in 2014. City officials plan to build a new multipurpose stadium. The projected cost of the stadium in 2014 dollars is $7.5 million. Assume that it is the end of 2011 and city officials intend to invest an equal amount of money at the end of each of the next three years in an account that will pay 8.75 percent. What is the annual payment necessary to meet this projected cost of the stadium?
Solution: 0
8.75%
1
2
3
├───────┼────────┼───────┤ PMT
PMT
PMT
FVAn = $7,500,000 Expected construction costs = FVA = $7,500,000 Return on investment = i = 8.75% Payment required to meet target = PMT Using the FVA equation: FVA n PMT
(1 i ) n 1 i
(1.0875)3 1 $7,500,000 PMT 0.0875 PMT
$7,500,000 $7,500,000 (1.0875) 3 1 3.2702 0.0875
$2, 293, 468.39
50
6.4
You have just won a lottery that promises an annual payment of $118,312 beginning immediately. You will receiver a total of 10 payments. If you can invest the cash flows in an investment paying 7.65 percent annually, what is the present value of this annuity?
Solution: 0
7.65% 1
2
3
9
10
├───────┼────────┼───────┼………………┼───────┤ $118,312
$118,312
$$118,312
$$118,312
$118,312
Annual payment = PMT = $118,312 No. of payments = n = 10 Required rate of return = 7.65% Type of annuity = Annuity due Present value of investment = PVA10
1 PVA n PMT
1 (1 i ) n (1 i ) i 1
$118,312
1 (1.0765)10 (1.0765) 0.0765
$118,312 6.8173 1.0765 $868, 272.71
6.5
Which of the following investments has the highest effective annual rate (EAR)? a. A bank CD that pays 5.50% interest quarterly b. A bank CD that pays 5.45% monthly c. A bank CD that pays 5.65% annually d. A bank CD that pays 5.55% semiannually e. A bank CD that pays 5.35% daily (on a 365-day basis)
Solution:
51 a.
Interest rate on CD = i = 5.50% Frequency of compounding = m = 4 Effective annual rate = EAR m1
4
i 0.055 EAR 1 1 1 1 4 m 1.05615 1 5.62%
b.
Interest rate on CD = i = 5.45% Frequency of compounding = m = 12 Effective annual rate = EAR m1
12
i 0.0545 EAR 1 1 1 1 12 m 1.055882 1 5.59%
c.
Interest rate on CD = i = 5.65% Frequency of compounding = m = 1 Effective annual rate = EAR m1
1
i 0.0565 EAR 1 1 1 1 1 m 1.0565 1 5.65%
d.
Interest rate on CD = i = 5.55% Frequency of compounding = m = 2 Effective annual rate = EAR m1
2
i 0.0555 EAR 1 1 1 1 2 m 1.05576 1 5.58%
e.
Interest rate on CD = i = 5.35% Frequency of compounding = m = 365 Effective annual rate = EAR
52 m1
i 0.0535 EAR 1 1 1 365 m 1.05495 1 5.5%
365
1
c. The bank CD that pays 5.65 percent annually has the highest yield.
53
Appendix: Deriving the Formula for the Present Value of an Ordinary Annuity In this chapter we showed that the formula for a perpetuity can be obtained from the formula for the present value of an ordinary annuity if n is set equal to ∞. It is also possible to go the other way. In other words, the present value of an ordinary annuity formula can be derived from the formula for a perpetuity. In fact, this is how the annuity formula was originally obtained. To see how this was done, assume that someone has offered to pay you $1 per year forever, beginning next year, but that, in return, you will have to pay that person $1 per year forever, beginning in year n + 1.
Solution: In this problem, you will receive an annual payment that grows at a rate of g forever. In return, you will have to pay that person $1(1 + g)n each year forever, beginning in year n + 1. The cash flows that you will receive can be represented as in the following time line. 0
1
$0
3
n-1 n
n+1 n+2
∞
$1(1+g) $1(1+g)2……$1(1+g)n-2 $1(1+g)n-1 $1(1+g)n
Receive $1 Pay
2
$0
$0
$0
$0
$1(1+g)n
The first time line shows the cash flows for the perpetuity that you will receive. This perpetuity is worth:
PVReceive
$1 CF i g (i g)
The second time line shows the cash flows for the perpetuity that you will pay. The present value of what you owe is:
PVOwe
$1 (1 g) n CF (1 g) n (i - g) (i - g) n (1 i ) n 1 i
Notice that if you subtract, year-by-year, the cash flows you would pay from the cash flows you would receive, you get the cash flows for an n-year annuity.
54 0
1
2
3
n–1 n
n+1 n+2
∞
Difference $1 $1(1 + g) $1(1 + g)2………$1(1 + g)n-2 $1(1 + g)n-1 $0 Therefore, the value of the offer equals the value of an n-year growing annuity. Solving for the difference between PVOwe from PVReceive ,we see that this is the same as Equation 6.5. PVRe ceive PVOwe
CF(1 g) n n CF CF 1 g i 1 (i - g) (1 i ) n (i - g) 1 i
Problem 6A.1
In the chapter text, you saw that the formula for a growing perpetuity can be obtained from the formula for the present value of a growing annuity if n is set equal to . It is also possible to go the other way. In other words, the present value of a growing annuity formula can be derived from the formula for a growing perpetuity. In fact, this is how Equation 6.5 was actually derived. Show how Equation 6.5 can be derived from Equation 6.6.
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