Accounting
FNAN 303
Solutions to test bank problems time value of money, part 2
Some answers may be slightly different than provided solutions due to rounding
1. Rainbow bought a new home entertainment system today from Andres Electronics. She will receive a cash rebate of $230 from Andres Electronics today, pay $720 to Andres Electronics in 1 year, receive a cash rebate of $1,640 from Andres Electronics in 2 years, and pay $4,980 to Andres Electronics in 4 years. If the discount rate is 18.94 percent, then what is the present value of the cash flows associated with this transaction? Note: the correct answer is less than zero. (Fall 2009, test 1, question 6) (Spring 2010, test 1, question 5)
(Fall 2010, test 1, question 8) (Fall 2011, test 1, question 7)
(Spring 2012, test 1, question 5) (Fall 2012, test 1, question 7)
(Spring 2015, test 1, question 4) (Fall 2015, final, question 3)
(Fall 2016, final, question 2) (Fall 2017, test 1, question 4)
(Spring 2018, final, question 2)
Time
0
1
2
3
4
Cash flows
230
-720
1,640
0
-4,980
Present value
?
PV = C0 + [C1/(1+r)1] + [C2/(1+r)2] + [C3/(1+r)3] + [C4/(1+r)4]
C0 = 230
C1 = -720
C2 = 1,640
C3 = 0
C4 = -4,980
r = .1894
PV = 230 + [-720/(1.1894)] + [1,640/(1.1894)2] + [0/(1.1894)3] + [-4,980/(1.1894)4]
= -$1,704.45
Answers may differ slightly due to rounding
2. Gloriana has an investment that is worth $5,000 and has an expected return of 19.9 percent. The investment is expected to pay her $7,000 in 3 years from today and X in 2 years from today. What is X?
A. An amount less than $0 or an amount equal to or greater than $1,600
B. An amount equal to or greater than $0 but less than $900
C. An amount equal to or greater than $900 but less than $1,200
D. An amount equal to or greater than $1,200 but less than $1,400
E. An amount equal to or greater than $1,400 but less than $1,600
(Fall 2010, test 1, question 4)
(Spring 2011, test 1, question 5)
(Spring 2011, test 2, question 1)
(Fall 2011, test 2, question 1)
(Spring 2013, test 1, question 5)
(Fall 2013, test 1, question 7)
(Spring 2014, test 1, question 5)
(Spring 2016, test 1, question 4)
Time
0
1
2
3
Expected cash flow
0
0
X
7,000
Present value
5,000
PV = C0 + [C1/(1+r)1] + [C2/(1+r)2] + [C3/(1+r)3]
In this case, PV = [C2/(1+r)2] + [C3/(1+r)3]
PV = 5,000
C3 = 7,000
C2 = X
r = .199
5,000 = [C2/(1.199)2] + [7,000/(1.199)3]
= [C2/(1.199)2] + 4,061.07
So 5,000 4,061.07 = [C2/(1.199)2]
= 938.93
= [C2/(1.199)2]
So [C2/(1.199)2] = 938.93
So C2 = X = 938.93 (1.199)2 = $1,349.81
Answer: D
$1,349.81 is an amount equal to or greater than $1,200 but less than $1,400
3. Almond just bought a new cracker. To pay for the cracker, the company took out a loan that requires Almond to pay the bank a special payment of $5,500 in 3 months and also pay the bank regular payments of $2,130 each month forever. The interest rate on the loan is 1.07 percent per month and the first monthly payment of $2,130 will be paid in 1 month. What was the price of the cracker?
(Fall 2014, test 1, question 5)
Timeline tip for FNAN 303: the given rate is for a month, so the timeline period is a month
From the timeline, we can see that the cash flows reflect a cash flow of -$5,500 in 3 months and a fixed perpetuity with a cash flow of -$2,130
Time
0
1
2
3
4
5
Regular pmt #
1
2
3
4
5
CF
-2,130
-2,130
-2,130 +
(-5,500)
-2,130
-2,130
Price of the cracker
= Opposite of the present value of all the cash flows associated with the loan
= PV of the cash flow of -$5,500 in 3 months + PV of the fixed perpetuity
PV of the cash flow of -$5,500 cash flow in 3 months
PV = -5,500 / (1.0107)3 = -5,327
Mode is not relevant, since PMT = 0
Enter 3 1.07 0 -5,500
N I% PV PMT FV
Solve for 5,327
PV of the perpetuity
PV = C/r
C = -$2,130
r = .0107
PV = -2,130/.0107 = -199,065
Price of the cracker
Price of the cracker
= Opposite of (PV of the cash flow of -$5,500 in 3 months + PV of the fixed perpetuity)
= Opposite of [-5,327 + (-199,065)]
= -[-5,327 + (-199,065)]
= -[-$204,392]
= $204,392
4. You own building A and building B. The next cash flow for each building is expected in 1 year. Building A has a cost of capital of 6.50 percent and is expected to produce annual cash flows of $341,000 forever. Building B is worth $5,200,000 and is expected to produce annual cash flows of $329,000 forever. Which assertion is true?
A. Building A is more valuable than building B and building A is more risky than building B
B. Building A is more valuable than building B and building B is more risky than building A
C. Building B is more valuable than building A and building A is more risky than building B
D. Building B is more valuable than building A and building B is more risky than building A
E. Building A and building B either have the same value, the same level of risk, or both the same value and level of risk.
(Spring 2011, test 1, question 6)
(Fall 2011, test 1, question 8)
Compare values
The cash flows for building A are a fixed perpetuity, so PV = C/r
C = $341,000 and r = .0650
PV(A) = $341,000 / .0650 = $5,246,154
PV(B) = $5,200,000 is given
PV(A) = $5,246,154 > $5,200,000 = PV(B), so building A is more valuable than building B
Compare risks
Recall that discount rate, cost of capital, opportunity cost of capital, and expected return all refer to the same concept, and they are all related to risk. The greater the risk of an asset, then the greater is the discount rate, cost of capital, opportunity cost of capital, and expected return associated with it.
Therefore, the building that is more risky is the one with the higher cost of capital
Cost of capital for building A = r(A) = 6.50% = .0650 is given
The cash flows for building B are a fixed perpetuity, so PV = C/r and r = C/PV
C = $329,000 and PV = 5,200,000
r(B) = $329,000 / 5,200,000 = .0633 = 6.33%
r(A) = .0650 > .0633 = r(B) so building A is more risky than building B
Putting it together:
A. Building A is more valuable than building B and building A is more risky than building B
5. An investment, which is worth $57,000 and has an expected return of 17.24 percent, is expected to pay fixed annual cash flows forever with the next annual cash flow expected in 1 year. What is the present value of the annual cash flow that is expected in 3 years from today?
(Spring 2014, test 1, question 6)
(Spring 2015, test 1, question 5)
(Spring 2016, test 1, question 5)
(Spring 2018, test 1, question 4)
Timeline tip for FNAN 303: the cash flows occur annually so the timeline period is a year
Time
0
1
2
3
4
Cash flow
$0
C
C
C
C
Present value of perpetuity
$57,000
Approach
1) Find the amount of the cash flow expected in 3 years from today
2) Find the present value of the cash flow expected in 3 years from today
1) Find the amount of the cash flow expected in 3 years from today
The cash flows reflect a fixed perpetuity, so the cash flow expected in 3 years is the same as the cash flow expected every year forever
PV = C / r and C = PV r
PV = $57,000
r = .1724
C = $57,000 .1724 = $9,826.80
2) Find the present value of the cash flow expected in 3 years from today
PV0 = C3 / (1 + r)3
The cash flow expected in 3 years = C3 = $9,826.80
r = .1724
PV0 = $9,826.80 / 1.17243
= $6,097.96
6. Lorenas Cart took out a loan from the bank today for $248,000. The loan requires Lorenas Cart to make a special payment of $86,000 to the bank in 4 years and also make regular, fixed payments of X to the bank each year forever. The interest rate on the loan is 8.91 percent per year and the first regular, fixed annual payment of X will be made to the bank in 1 year. What is X, the amount of the regular, fixed annual payment?
(Fall 2009, test 1, question 5 simpler version on exam than in test bank)
(Fall 2011, test 1, question 9)
(Spring 2013, test 1, question 6)
(Fall 2013, test 1, question 8)
(Spring 2015, final, question 4)
(Fall 2015, test 1, question 4)
(Spring 2017, final, question 2)
(Fall 2017, final, question 2)
From the timeline, we can see that the cash flows reflect a cash flow of -$86,000 in 4 years and a fixed perpetuity
Time
0
1
2
3
4
5
6
Regular pmt #
1
2
3
4
5
6
CF
X
X
X
X + (-86,000)
X
X
PV of all CFs
248,000
Loan amount = opposite of (PV of the cash flow of -$86,000 in 4 years + PV of the fixed perpetuity)
Approach
1) Find the PV of the cash flow of -$86,000 in 4 years
2) Find the PV of the perpetuity
3) Find the regular, fixed cash flow associated with perpetuity
1) PV of the cash flow of -$86,000 in 4 years
PV = -86,000 / (1.0891)4 = -61,126
2) Find the PV of the perpetuity
-Loan amount = PV of the cash flow of -$86,000 in 4 years + PV of the fixed perpetuity
So -248,000 = -61,126 + PV of the fixed perpetuity
So -248,000 + 61,126 = -186,874 = PV of the fixed perpetuity
3) Find the regular, fixed cash flow associated with perpetuity
PV = C/r so C = PV r
PV = -186,874
r = .0891
C = -186,874 .0891 = -16,650
Answers may differ due to rounding
The regular, fixed cash flow associated with the perpetuity is -$16,650 per year, so the annual payment is $16,650
7. Bob has an investment worth $307,000. The investment will make a special payment of X to Bob in 2 months and the investment also will make regular, fixed monthly payments of $2,390 to Bob forever. The expected return for the investment is 1.13 percent per month and the first regular, fixed monthly payment of $2,390 will be made to Bob in 1 month. What is X, the amount of the special payment that will be made to Bob in 2 months?
(Spring 2012, test 1, question 6)
(Spring 2013, final, question 2)
(Fall 2013, test 2, question 1)
(Spring 2017, test 1, question 4)
Timeline tip for FNAN 303: the given rate is for a month, so the timeline period is a month
From the timeline, we can see that the cash flows reflect a perpetuity with regular, monthly cash flows of $2,390 and an extra cash flow of X in 2 months
Time
0
1
2
3
4
5
Regular pmt #
1
2
3
4
5
CF
$2,390
$2,390 + X
$2,390
$2,390
$2,390
Investment value = PV of the special cash flow made in 2 months + PV of the fixed perpetuity
Approach
1) Find the PV of the perpetuity
2) Find the PV of the special cash flow made in 2 months
3) Find the amount of the special cash flow in 2 months
1) PV of the perpetuity
PV = C/R
C = 2,390
R = .0113
PV (perpetuity) = 2,390 / .0113 = $211,504
2) Find the PV of the special cash flow made in 2 months
Investment value = PV of the special cash flow made in 2 months + PV of the fixed perpetuity
So 307,000 = PV of the special cash flow made in 2 months + 211,504
So 307,000 211,504 = PV of the special cash flow made in 2 months
= 95,496
3) Find the amount of the special cash flow in 2 months
PV = C2 / (1 + R)2 = X / (1 + R)2
So 95,496 = X / (1.0113)2
So X = 95,496 (1.0113)2
= $97,666
8. An investment is expected to generate annual cash flows forever. The first annual cash flow is expected in 1 year and all subsequent annual cash flows are expected to grow at a constant rate annually. The cash flow expected in 2 years from today is expected to be $7,300 and the cash flow expected in 6 years from today is expected to be $8,400. What is the cash flow expected to be in 5 years from today?
A. An amount less than $7,000 or an amount equal to or greater than $8,600
B. An amount equal to or greater than $7,000 but less than $7,400
C. An amount equal to or greater than $7,400 but less than $7,800
D. An amount equal to or greater than $7,800 but less than $8,200
E. An amount equal to or greater than $8,200 but less than $8,600
(Fall 2012, final, question 4) (Fall 2012, test 1, question 9)
(Fall 2013, final, question 3) (Spring 2014, test 1, question 7)
(Spring 2016, test 1, question 6)
Approach:
1) Find the annual growth rate
2) Find the cash flow expected in 5 years
Time
0
1
2
3
4
5
6
CF
0
C1
C2
C3
C4
C5
C6
CF
0
C1
7,300
C3
C4
C5
8,400
CF
0
C1
C2
C2 (1+g)
C2 (1+g)2
C2 (1+g)3
C2 (1+g)4
CF
0
C1
C2
C2 (1+g)
C2 (1+g)2
C2 (1+g)3
C6
CF
0
C1
C2
C3
C4
C5
C5 (1+g)
1) Find the annual growth rate
We know that Cb = Ca (1+g)(b-a) so C6 = C2 (1+g)(6-2) = C2 (1+g)4
C2 = 7,300
C6 = 8,400
So 8,400 = 7,300 (1+g)4
(8,400/7,300) = (1+g)4
(8,400/7,300)(1/4) = [(1+g)4](1/4) = 1 + g
= 1.0357
So g = 1.0357 1 = .0357
2) Find the cash flow expected in 5 years
Students should be comfortable finding the expected cash flow using 1) the growth rate and a known expected cash flow that takes place before the desired one and 2) the growth rate and a known expected cash flow that takes place after the desired one.
1) The growth rate and a known expected cash flow that takes place before the desired one
We know that Cb = Ca (1+g)(b-a) so C5 = C2 (1+g)(5-2) = C2 (1+g)3
C2 = 7,300 and g = .0357
So C5 = C2 (1+g)(5-2) = 7,300 (1.0357)3 = $8,110
2) The growth rate and a known expected cash flow that takes place after the desired one
We know that Cb = Ca (1+g)(b-a) so C6 = C5 (1+g)(6-5) = C5 (1+g)1
So 8,400 = C5 1.0357
C5 = 8,400 / 1.0357 = 8,110
Answer: D
$8,110 is an amount equal to or greater than $7,800 but less than $8,200
9. An investment, which has an expected return of 16.1%, is expected to make annual cash flows forever. The first annual cash flow is expected in 1 year and all subsequent annual cash flows are expected to grow at a constant rate of 3.7% per year. The cash flow in 1 year is expected to be $42,000. What is the present value (as of today) of the cash flow that is expected to be made in 5 years?
(Fall 2014, test 1, question 6)
(Fall 2015, final, question 4)
(Spring 2017, test 1, question 5)
Time
0
1
2
3
4
5
Cash flow
0
C1
C1 (1.037)
C1 (1.037)2
C1 (1.037)3
C1 (1.037)4
Cash flow
0
42,000
42,000 (1.037)
42,000 (1.037)2
42,000 (1.037)3
42,000 (1.037)4
Approach:
1) find the expected cash flow in 5 years
2) find the present value of the expected cash flow in 5 years
1) find the expected cash flow in 5 years
We know that C5 = C1 (1+g)4
C1 = 42,000
g = .037
C5 = 42,000 1.0374
= $48,570
Time
0
1
2
3
4
5
Cash flow
0
C1
C1 (1.037)
C1 (1.037)2
C1 (1.037)3
C1 (1.037)4
Cash flow
0
42,000
42,000 (1.037)
42,000 (1.037)2
42,000 (1.037)3
48,570
Present value
PV0
2) find the present value of the expected cash flow in 5 years
We know that PV0 = C5 / (1+r)5
C5 = 48,570
r = .161
PV0 = 48,570 / (1.161)5
= $23,025
10. Hazelnut just bought a new cracker. To pay for the cracker, the company took out a loan that requires Hazelnut to pay the bank a special payment of $8,160 in 1 month and also pay the bank regular payments. The first regular payment is expected to be $1,240 in 1 month and all subsequent regular payments are expected to increase by 0.25 percent per month forever. The interest rate on the loan is 0.98 percent per month. What was the price of the cracker?
Timeline tip for FNAN 303: the given rate is for a month, so the timeline period is a month
From the timeline, we can see that the cash flows reflect a cash flow of -$8,160 in 1 month and a growing perpetuity with an initial cash flow of -$1,240 and a growth rate of .0025
Time
0
1
2
3
4
5
Regular pmt #
1
2
3
4
5
CF
-$1,240 + (-$8,160)
-$1,240 1.0025
-$1,240 1.00252
-$1,240 1.00253
-$1,240 1.00254
Price of the cracker
= Opposite of the present value of all the cash flows associated with the loan
= Opposite of (PV of the cash flow of -$8,160 in 1 month + PV of the growing perpetuity)
PV of the cash flow of -$8,160 cash flow in 1 month
PV = -8,160 / (1.0098)1 = -8,081
Mode is not relevant, since PMT = 0
Enter 1 0.98 0 -8,160
N I% PV PMT FV
Solve for 8,081
PV of the perpetuity
PV = C1/(r g)
C1 = -$1,240
r = .0098
g = .0025
PV = -1,240 / (.0098 .0025)
= -1,240 / .0073
= -169,863
Price of the cracker
Price of the cracker
= Opposite of (PV of the cash flow of -$8,160 in 1 month + PV of the growing perpetuity)
= Opposite of [-8,081 + (-169,863)]
= -[-8,081 + (-169,863)]
= -[-$177,944]
= $177,944
11. A dry cleaning store is expected to make annual cash flows forever. The cost of capital for the store is 14.20 percent. The next annual cash flow is expected in one year from today and all subsequent cash flows are expected to grow annually by 3.40 percent. What is the value of the dry cleaning store if the cash flow in 4 years from today is expected to be $241,000?
(Spring 2012, test 1, question 7)
(Spring 2013, test 1, question 8)
(Spring 2013, final, question 3)
(Spring 2014, final, question 3)
(Fall 2016, test 1, question 4)
Time
0
1
2
3
4
Cash flow
0
C1
C2
C3
C4
Cash flow
0
C1
C1 (1.034)
C1 (1.034)2
C1 (1.034)3
Cash flow
0
C1
C1 (1.034)
C1 (1.034)2
$241,000
Present value
?
PV = C1 / (r g)
We know r and g, so if we find C1, we can get PV
Approach:
1) Find the expected cash flow in 1 year
2) Use the expected cash flow in 1 year to find the value today
1) Find the expected cash flow in 1 year
The cash flows reflect a growing perpetuity
We know that C4 = C1 (1+g)3
C4 = 241,000
g = .034
241,000 = C1 (1.034)3
So C1 = 241,000 / (1.034)3
= $217,999
2) Use the expected cash flow in 1 year to find the value today
The cash flows reflect a growing perpetuity
We know that PV = C1 / (r g)
C1 = $217,999
r = .142
g = .034
PV = 217,999 / (.142 .034)
= 217,999 / .108
= $2,018,509
12. Takashi has an investment worth $237,000. The investment will make a special payment of X to Takashi in 4 quarters in addition to making regular quarterly payments to Takashi forever. The first regular quarterly payment to Takashi is expected to be $1,870 and will be made in 3 months. All subsequent regular quarterly payments are expected to increase by 0.29 percent per quarter forever. The expected return for the investment is 1.37 percent per quarter. What is X, the amount of the special payment that will be made to Takashi in 4 quarters?
(Spring 2012, test 2, question 1)
(Fall 2017, final, question 3)
(Spring 2018, test 1, question 5)
Timeline tip for FNAN 303: the given rate is for a quarter, so the timeline period is a quarter
From the timeline, we can see that the cash flows reflect a growing perpetuity with its next cash flow of $1,870 in one quarter, a growth rate of 0.29%, and an extra cash flow of C in 4 quarters
Time
0
1
2
3
4
5
Regular pmt #
1
2
3
4
5
CF
$1,870
$1,870 1.0029
$1,870 1.00292
($1,870 1.0029)3
+ X
$1,870 1.00294
Investment value = PV of the special cash flow made in 4 quarters + PV of the growing perpetuity
Approach
1) Find the PV of the perpetuity
2) Find the PV of the special cash flow made in 4 quarters
3) Find the amount of the cash flow made in 4 quarters
1) PV of the perpetuity
PV = C1/ (R g)
C1 = 1,870
R = .0137
g = .0029
PV (perpetuity) = 1,870 / (.0137 .0029)
= 1,870 / .0108
= $173,148
2) Find the PV of the special cash flow made in 4 quarters
Investment value = PV of the special cash flow made in 4 quarters + PV of the growing perpetuity
So 237,000 = PV of the special cash flow made in 4 quarters + 173,148
So 237,000 173,148 = PV of the special cash flow made in 4 quarters
= 63,852
3) Find the amount of the cash flow made in 4 quarters
PV = C4 / (1 + R)4 = X / (1 + R)4
So 63,852 = C4 / (1.0137)4
So C4 = 63,852 (1.0137)4
= $67,424
Answers may differ slightly due to rounding
13. Pistachio just bought a new cracker for $300,000. To pay for the cracker, the company took out a loan that requires Pistachio to pay the bank a special payment of $55,000 in 2 months and also make regular monthly payments forever. The first regular payment is expected in 1 month and all subsequent regular payments are expected to increase by 0.2 percent per month forever. The interest rate on the loan is 1.0 percent per month. What is the payment expected to be in 1 month?
(Spring 2010, test 1, question 7)
(Fall 2012, test 1, question 10)
Timeline tip for FNAN 303: the given rate is for a month, so the timeline period is a month
From the timeline, we can see that the cash flows reflect a cash flow of -$55,000 in 2 months and a growing perpetuity with a growth rate of .002
Time
0
1
2
3
4
5
Regular pmt #
1
2
3
4
5
CF
C1
($55,000) + (C1 1.002)
C1 1.0022
C1 1.0023
C1 1.0024
Price of the cracker
= Opposite of the present value of all the cash flows associated with the loan
= Opposite of (PV of the cash flow of -$55,000 in 2 months + PV of the growing perpetuity)
Approach
1) Find the PV of the cash flow of -$55,000 cash flow in 2 months
2) Find the PV of the perpetuity
3) Find the cash flow expected in 1 month
1) PV of the cash flow of -$55,000 cash flow in 2 months
PV = -55,000 / (1.010)2 = -53,916
2) Find the PV of the perpetuity
Price of the cracker
= Opposite of (PV of the cash flow of -$55,000 in 2 months + PV of the growing perpetuity)
So 300,000 = -(-53,916 + PV of the growing perpetuity)
= 53,916 PV of the growing perpetuity
So 300,000 53,916 = 246,084 = -PV of the growing perpetuity
So PV of the growing perpetuity = -246,084
3) Find the cash flow expected in 1 month
PV = C1/(r g) so C1 = PV (r g)
PV = -246,084
r = .010
g = .002
(r g) = .010 .002 = .008
C1 = -246,084 .008 = -1,968.67
The cash flow expected in 1 month is -$1,968.67, so the payment expected in 1 month is $1,968.67
14. You own two investments, A and B, that have a combined total value of $24,300. Investment A is expected to make its next payment in 1 month. As next payment is expected to be $112 and subsequent payments are expected to grow by 0.23 percent per month forever. The expected return for investment A is 1.04 percent per month. Investment B is expected to pay $182 each quarter forever and the next payment is expected in 3 months. What is the quarterly expected return for investment B?
(Fall 2009, test 1, question 4)
(Fall 2009, final, question 1)
(Fall 2010, test 1, question 10)
(Fall 2017, test 1, question 5)
To solve
1) Find the value of investment A
2) Find the value of investment B as the combined value of both A and B minus the value of A
3) Find the expected return for B
1) Find the value of investment A
For a growing perpetuity, PV = C1/(r g) and r = (C1/PV) + g
C1 = $112, r = .0104, and g = .0023
PV = $112 / (.0104 .0023) = $112 / .0081 = $13,827
Investment A is worth $13,827
2) Find the value of investment B as the combined value of both A and B minus the value of A
Value of B = value of A and B value of A
Value of A and B = $24,300
Value of A = $13,827
Value of B = $24,300 $13,827 = $10,473
3) Find the expected return for B
For a fixed perpetuity, PV = C/r and r = C/PV
C = $182
PV = $10,473
r = 182/10,473 = 0.0174 = 1.74%
The expected return for investment B is 1.74 percent per quarter
15. Heather owns a roller rink that is worth $760,000 and is expected to make annual cash flows forever. The cost of capital for the roller rink is 8.8%. The next annual cash flow is expected in one year from today and all subsequent cash flows are expected to grow annually by 2.3%. What is the cash flow produced by the roller rink in 5 years from today expected to be?
(Fall 2010, test 1, question 9)
(Fall 2010, final, question 3)
(Fall 2011, test 1, question 10)
(Spring 2012, final, question 2)
(Fall 2013, test 1, question 10)
(Fall 2014, final, question 3)
Time
0
1
2
3
4
5
Cash flow
0
C1
C2
C3
C4
C5
Cash flow
0
C1
C1 (1.023)
C1 (1.023)2
C1 (1.023)3
C1 (1.023)4
Present value
760,000
Approach:
1) find the expected cash flow in 1 year
2) use the expected cash flow in 1 year and the growth rate to find the expected cash flow in 5 years
1) find the expected cash flow in 1 year
The cash flows reflect a growing perpetuity
PV = C1 / (r g) and C1 = PV (r g)
PV = 760,000
r = .088
g = .023
C1 = 760,000 (.088 .023)
= 760,000 .065
= $49,400
2) use the expected cash flow in 1 year and the growth rate to find the expected cash flow in 5 years
We know that C5 = C1 (1+g)4
= 49,400 (1.023)4
= 54,104
16. Walnut just bought a new cracker for $300,000. To pay for the cracker, the company took out a loan that requires Walnut to pay the bank a special payment of $55,000 in 2 months and also make regular monthly payments forever. The first regular payment is expected in 1 month and is expected to be $1,800. All subsequent regular payments are expected to increase by a constant rate each month forever. The interest rate on the loan is 1.00 percent per month. What is the monthly growth rate of the regular payments expected to be?
(Spring 2013, test 1, question 7)
(Spring 2014, test 1, question 8 similar, but an investment with no extra cash flow)
(Spring 2015, test 1, question 7)
Timeline tip for FNAN 303: the given rate is for a month, so the timeline period is a month
From the timeline, we can see that the cash flows reflect a cash flow of -$55,000 in 2 months and a growing perpetuity with an initial cash flow of -$1,800 and a growth rate of g
Time
0
1
2
3
4
5
Regular pmt #
1
2
3
4
5
CF
-1,800
(55,000) +
(-1,800 (1 + g))
-1,800
(1 + g)2
-1,800
(1 + g)3
-1,800
(1 + g)4
Price of the cracker
= Opposite of the present value of all the cash flows associated with the loan
= Opposite of (PV of the cash flow of -$55,000 in 2 months + PV of the growing perpetuity)
Approach
1) Find the PV of the cash flow of -$55,000 cash flow in 2 months
2) Find the PV of the perpetuity
3) Find the growth rate associated with perpetuity
1) PV of the cash flow of -$55,000 cash flow in 2 months
PV = -55,000 / (1.0100)2 = -53,916
2) Find the PV of the perpetuity
Price of the cracker
= Opposite of (PV of the cash flow of -$55,000 in 2 months + PV of the growing perpetuity)
So 300,000 = -(-53,916 + PV of the growing perpetuity)
= 53,916 PV of the growing perpetuity
So 300,000 53,916 = 246,084 = -PV of the growing perpetuity
So PV of the growing perpetuity = -246,084
3) Find the regular, growing cash flow associated with perpetuity
PV = C1/(r g) so r = (C1 / PV) + g so g = r (C1 / PV)
PV = -246,084
r = .0100
C1 = -1,800
g = .0100 (-1,800/ -246,084)
= .0100 .0073 = .0027 = 0.27 percent per month
The growth rate of the regular payment is expected to be 0.27 percent per month
17. Shagaf owns a laser tag center that is worth $560,000 and is expected to make annual cash flows forever. The cost of capital for the laser tag center is 9.7%. The next annual cash flow is expected in 1 year and is expected to be $45,920. All subsequent cash flows are expected to grow annually at a constant growth rate. What is the cash flow produced by the laser tag center in 4 years expected to be?
(Spring 2011, test 1, question 7)
(Spring 2011, final, question 3)
Time
0
1
2
3
4
Cash flow
0
C1
C2
C3
C4
Cash flow
0
45,920
45,920 (1+g)
45,920 (1+g)2
45,920 (1+g)3
Present value
560,000
Approach:
1) find the expected growth rate
2) use the expected cash flow in 1 year and the growth rate to find the expected cash flow in 4 years
1) find the expected growth rate
The cash flows reflect a growing perpetuity
PV = C1 / (r g) and g = r (C1 / PV)
C1 = 45,920
PV = 560,000
r = .097
g = r (C1 / PV)
= .097 (45,920 / 560,000)
= .097 .082 = .015
2) use the expected cash flow in 1 year and the growth rate to find the expected cash flow in 4 years
We know that C4 = C1 (1+g)3
= 45,920 (1.015)3
= $48,018
18. You own two investments, A and B, that have a combined total value of $32,500. Investment A is expected to pay $850 per year forever; its next payment is expected in 1 year; and its expected return is 6.16 percent per year. Investment B is also expected to make annual payments forever and make its next payment in 1 year. Investment Bs next payment is expected to be $975 and all subsequent payments are expected to grow by 0.73 percent per year forever. What is the annual expected return for investment B?
To solve
1) Find the value of investment A
2) Find the value of investment B as the combined value of both A and B minus the value of A
3) Find the expected return for B
1) Find the value of investment A
For a fixed perpetuity, PV = C/r and r = C/PV
C = $850 and r = .0616
PV = $850 / .0616 = 13,799
Investment A is worth $13,799
2) Find the value of investment B as the combined value of both A and B minus the value of A
Value of B = value of A and B value of A
Value of A and B = $32,500
Value of A = $13,799
Value of B = $32,500 $13,799 = $18,701
3) Find the expected return for B
For a growing perpetuity, PV = C1/(r g) and r = (C1/PV) + g
C1 = $975
PV = $18,701
g = .0073
r = (975/18,701) + .0073
= 0.0521 + .0073 =
.0594 = 5.94%
19. What is the value of an investment that will pay investors $1,850 per month for 7 months and will also pay investors an additional $6,400 in 1 month from today if the expected return for the investment is 1.27% per month and the first $1,850 monthly payment will be paid to investors in one month from today?
(Fall 2010, test 2, question 1)
(Spring 2012, test 1, question 8)
(Spring 2016, test 1, question 7)
(Spring 2018, test 1, question 6)
From the timeline, we can see that the cash flows reflect a cash flow of $6,400 in 1 month and a 7-period ordinary annuity with a payment of $1,850
Time
0
1
2
3
4
5
6
7
Regular pmt #
1
2
3
4
5
6
7
CF
1,850 +
6,400
1,850
1,850
1,850
1,850
1,850
1,850
Value of investment = PV of the cash flow of $6,400 in 1 month + PV of the 7-period annuity
PV of the cash flow of $6,400 cash flow in 1 month
PV0 = 6,400 / (1.0127)1 = 6,320
Mode is not relevant, since PMT = 0
Enter 1 1.27 0 6,400
N I% PV PMT FV
Solve for -6,320
PV of the 7-period annuity
END Mode
Enter 7 1.27 1,850 0
N I% PV PMT FV
Solve for -12,316
Value of investment
The value of the investment = 6,320 + 12,316 = $18,636
20. A diamond mine
