Accounting

(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

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)

= -[-5,327 + (-199,065)]

= $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

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. Lorena’s Cart took out a loan from the bank today for $248,000. The loan requires Lorena’s 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

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

(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

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

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)

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

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

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)

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

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

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. A’s 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)

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

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

(Spring 2011, test 1, question 7)

(Spring 2011, final, question 3)

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 B’s 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

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 is expected to produce regular annual cash flows of $1 million for 8 years with the first regular cash flow expected in 1 year from today. In addition to the regular cash flows of $1,000,000, the diamond mine is also expected to produce an extra cash flow of $3,000,000 in 8 years from today. The cost of capital for the mine is 12 percent. What is the value of the mine?

From the timeline, we can see that the cash flows reflect an 8-period annuity with payments of $1,000,000 plus an extra payment in 8 years of $3,000,000

N = 8 since there are “annual cash flows of $1 million for 8 years ”

Value of investment = PV of the 8-period annuity + PV of the cash flow of $3,000,000 in 8 years

PV of the 8-period annuity

END Mode

Enter 8 12 1,000,000 0

N I% PV PMT FV

Solve for -4,967,640

PV of the cash flow of $3,000,000 in 8 years

PV = C8 / (1+r)8

r = .12

C8 = 3,000,000 (note: this reflects the cash flow in 8 years not associated with the annuity)

PV = 3,000,000 / (1.12)8 = 1,211,650

Mode is not relevant, since PMT = 0

Enter 8 12 0 3,000,000

N I% PV PMT FV

Solve for -1,211,650

Aggregate the two sources of value

The value of the investment = 4,967,640 + 1,211,650 = $6,179,290

Important note

You can do this problem in one calculator step: if you input in $3,000,000 as FV, the calculator will treat that cash flow as taking place in N periods, which would be in 8 periods. The $3,000,000 is expected in 8 years, so the answer would be correct.

END Mode

Enter 8 12 1,000,000 3,000,000

N I% PV PMT FV

Solve for -6,179,289

21. Bob has an investment worth $300,000. The investment will make a special payment of X to Bob in 2 years from today. The investment also will make regular, fixed annual payments of $65,000 to Bob with the first of these payments made to Bob in 1 year from today and the last of these annual payments made to Bob in 6 years from today. The expected return for the investment is 10 percent per year. What is X, the amount of the special payment that will be made to Bob in 2 years?

(Fall 2014, test 1, question 9)

Timeline tip for FNAN 303: the given rate is for a year, so the timeline period is a year

From the timeline, we can see that the cash flows reflect an annuity with regular, annual cash flows of $65,000 and an extra cash flow of X in 2 years

Investment value = PV of the special cash flow made in 2 years + PV of the ordinary annuity

Approach

1) Find the PV of the annuity

2) Find the PV of the special cash flow made in 2 years

3) Find the amount of the cash flow made in 2 years

1) PV of the annuity

END mode

Enter 6 10 65,000 0

N I% PV PMT FV

Solve for -283,092

PV of the annuity = 283,092

2) Find the PV of the special cash flow made in 2 years

Investment value = PV of the special cash flow made in 2 years + PV of the ordinary annuity

So 300,000 = PV of the special cash flow made in 2 years + 283,092

So 300,000 – 283,092 = PV of the special cash flow made in 2 years

= 16,908

3) Find the amount of the cash flow made in 2 years

PV = C2 / (1 + R)2 = X / (1 + R)2

So 16,908 = X / (1.10)2

So X = 16,908 × (1.10)2

= $20,459

22. Bianca took out a loan from the bank today for X. She plans to repay this loan by making payments of $850 per month for a certain amount of time. If the interest rate on the loan is 0.97 percent per month, she makes her first $850 payment later today, and she makes her final monthly payment of $850 in 7 months, then what is X, the amount of the loan?

(Fall 2012, test 2, question 2)

(Spring 2013, test 1, question 9)

(Fall 2013, final, question 4)

(Spring 2014, test 1, question 9)

(Spring 2015, final, question 3)

(Fall 2015, test 1, question 6)

This is a problem where we need to find the present value of an annuity due. We have a series of fixed payments that start immediately and a discount rate and we want to know what the value of those payments is. The size of a loan is the opposite of the present value of the cash flows associated with the loan payments.

BEGIN mode

Enter 8 0.97 -850 0

N I% PV PMT FV

Solve for 6,576

Bianca took out a loan for $6,576

23. Gil owns a ropes course that has a cost of capital of 12.0 percent. The ropes course is expected to produce annual cash flows of $50,000 for 6 years. The first annual cash flow of $50,000 is expected later today. In addition to the annual cash flows of $50,000, the ropes course is also expected to produce a special, one-time cash flow of $120,000 in 2 years from today. How much is Gil’s ropes course worth?

There are 6 cash flows of $50,000, because annual cash flows of $50,000 are expected “ for 6 years ”

Gil’s ropes course is worth the sum of the following:

1) The present value of a 6-period annuity due with payments of $50,000 and a discount rate of 12.0%.

2) A cash flow of $120,000 in 2 years with a discount rate of 12.0%

1) The PV of the annuity due

BEGIN mode

Enter 6 12 50,000 0

N I% PV PMT FV

Solve for -230,239

2) PV of $120,000 in 2 years

Note that mode is not relevant

Enter 2 12 0 120,000

N I% PV PMT FV

Solve for -95,663

Combine the 2 pieces

The ropes course is worth $230,239 + $95,663 = $325,902

(Answers may differ slightly due to rounding)

24. A diamond mine is expected to produce regular annual cash flows of $1 million with the first regular cash flow expected later today and the last regular cash flow expected in 7 years from today. In addition to the regular cash flows of $1,000,000, the diamond mine is also expected to produce an extra cash flow of $3,000,000 in 7 years from today. The cost of capital for the diamond mine is 12 percent. What is the value of the diamond mine?

From the timeline, we can see that the cash flows reflect an 8-period annuity due with payments of $1,000,000 plus an extra payment in 7 years of $3,000,000. Even though the last regular cash flow is expected in 7 years, it would be the 8th regular payment.

Value of investment = PV of the 8-period annuity due + PV of the cash flow of $3,000,000 in 7 years

PV of the 8-period annuity due

BEGIN Mode

Enter 8 12 1,000,000 0

N I% PV PMT FV

Solve for -5,563,756.54

PV of the cash flow of $3,000,000 in 7 years

PV = C7 / (1+r)7

r = .12

C7 = 3,000,000 (note: this reflects the cash flow in 7 years not associated with the annuity due)

PV = 3,000,000 / (1.12)7 = 1,357,047.65

Mode is not relevant, since PMT = 0

Enter 7 12 0 3,000,000

N I% PV PMT FV

Solve for -1,357,047.65

Aggregate the two sources of value

The value of the investment = 5,563,756.54 + 1,357,047.65 = $6,920,804.19

Important note

You can not do this problem in one calculator step: if you input in $3,000,000 as FV, the calculator will treat that cash flow as taking place in N periods from time 0, which would be in 8 periods. The $3,000,000 is expected in 7 years, so the answer would be incorrect.

A problem like this that you can not do in one step makes for an excellent exam problem, because it requires an understanding of the timeline associated with a problem and of what a calculator can and can not do. 25. A diamond mine is expected to produce regular annual cash flows of $1 million with the first regular cash flow expected later today and the last regular cash flow expected in 7 years from today. In addition to the regular cash flows of $1,000,000, the diamond mine is also expected to produce an extra cash flow of $3,000,000 in 8 years from today. The cost of capital for the diamond mine is 12 percent. What is the value of the diamond mine?

From the timeline, we can see that the cash flows reflect an 8-period annuity due with payments of $1,000,000 plus an extra payment in 8 years of $3,000,000. Even though the last regular cash flow is expected in 7 years, it would be the 8th regular payment.

Value of investment = PV of the 8-period annuity due + PV of the cash flow of $3,000,000 in 8 years

PV of the 8-period annuity due

BEGIN Mode

Enter 8 12 1,000,000 0

N I% PV PMT FV

Solve for -5,563,757

PV of the cash flow of $3,000,000 in 8 years

PV = C8 / (1+r)8

r = .12

C8 = 3,000,000 (note: this reflects the cash flow in 8 years not associated with the annuity due)

PV = 3,000,000 / (1.12)8 = 1,211,650

Mode is not relevant, since PMT = 0

Enter 8 12 0 3,000,000

N I% PV PMT FV

Solve for -1,211,650

Aggregate the two sources of value

The value of the investment = 5,563,757 + 1,211,650 = $6,775,407

Important note

Yes, you can do this problem in one calculator step: if you input in $3,000,000 as FV, the calculator will treat that cash flow as taking place in N periods from today, which would be in 8 periods. The $3,000,000 is expected in 8 years, so the answer would be correct.

BEGIN Mode

Enter 8 12 1,000,000 3,000,000

N I% PV PMT FV

Solve for -6,775,406

One-step and two-step answers may differ slightly due to rounding

26. Bob has an investment worth $412,000. The investment will make a special payment of X to Bob in 2 years from today. The investment also will make regular, fixed annual payments of $65,000 to Bob with the first of these payments made to Bob later today and the last of these annual payments made to Bob in 6 years from today. The expected return for the investment is 10.1 percent per year. What is X, the amount of the special payment that will be made to Bob in 2 years?

(Spring 2015, test 1, question 8)

(Spring 2016, final, question 2)

(Fall 2017, test 1, question 6)

Timeline tip for FNAN 303: the given rate is for a year, so the timeline period is a year

From the timeline, we can see that the cash flows reflect an annuity due with regular, annual cash flows of $65,000 and an extra cash flow of X in 2 years

Investment value = PV of the special cash flow made in 2 years + PV of the annuity due

Approach

1) Find the PV of the annuity due

2) Find the PV of the special cash flow made in 2 years

3) Find the amount of the cash flow made in 2 years

1) PV of the annuity due

BEGIN mode

Enter 7 10.1 65,000 0

N I% PV PMT FV

Solve for -347,264

PV of the annuity due = 347,264

2) Find the PV of the special cash flow made in 2 years

Investment value = PV of the special cash flow made in 2 years + PV of the annuity due

So 412,000 = PV of the special cash flow made in 2 years + 347,264

So 412,000 – 347,264 = PV of the special cash flow made in 2 years

= 64,736

3) Find the amount of the cash flow made in 2 years

PV = C2 / (1 + R)2 = X / (1 + R)2

So 64,736 = C2 / (1.101)2

So C2 = 64,736 × (1.101)2

= $78,473

27. Investment A will make N annual payments of $500 with the first of the N payments due immediately. Investment A has a value of $20,000. Investment B is an ordinary annuity that will make (N minus 1) annual payments of $500 with the first payment due in one year from today. If investment A and investment B have the same expected return, then what is the value of investment B?

A. $19,500

B. $20,000

C. $20,500

D. The value of investment B can be determined from the information given, but it is not equal to $19,500, $20,000, or $20,500

E. The value of investment B can not be determined from the information given

(Fall 2011, test 2, question 2)

(Fall 2012, final, question 5)

(Fall 2014, test 1, question 10)

(Spring 2017, test 1, question 7)

If A will make N annual payments with the first one immediately, then the Nth annual payment will be in N-1 years. Investment A is an annuity due.

A has a present value of $20,000

If B is an ordinary annuity that will make (N minus 1) annual payments of $500 with the first annual payment made in one year from today, then B will make N-1 annual payments with the first one in 1 year and the last one, which will be the (N-1)th in N-1 years.

We want to know the present value of B

Investment B makes all the same cash flows as investment A except that investment B does not pay $500 today.

The PV of A is $20,000 and the value of a $500 cash flow today is $500

Therefore, the PV of B is $20,000 – $500 = $19,500

Answer: A. $19,500

28. Investment A will make N annual payments of $500 with the first of the N payments due immediately. Investment B is an ordinary annuity that will make (N minus 1) annual payments of $500 with the first payment due in one year from today. Investment B has a value of $20,000. If investment A and investment B have the same expected return, then what is the value of investment A?

A. $19,500

B. $20,000

C. $20,500

D. The value of investment A can be determined from the information given, but it is not equal to $19,500, $20,000, or $20,500

E. The value of investment A can not be determined from the information given

(Fall 2011, test 2, question 2)

(Fall 2012, final, question 5)

(Fall 2014, test 1, question 10)

(Spring 2017, test 1, question 7)

If A will make N annual payments with the first one immediately, then the Nth annual payment will be in N-1 years. Investment A is an annuity due.

We want to know the present value of A

If B is an ordinary annuity that will make (N minus 1) annual payments of $500 with the first annual payment made in one year from today, then B will make N-1 annual payments with the first one in 1 year and the last one, which will be the (N-1)th in N-1 years.

B has a present value of $20,000

Investment B makes all the same cash flows as investment A except that investment B does not pay $500 today.

The PV of B is $20,000 and the value of a $500 cash flow today is $500

Therefore, the PV of A is $20,000 + $500 = $20,500

Answer: C. $20,500

29. Parhot just took out a loan from the bank for $3,600. He plans to repay this loan by making a special payment to the bank of $600 in 3 years and by also making equal, regular annual payments of X for 7 years. If the interest rate on the loan is 9.6 percent per year and he makes his first regular annual payment in 1 year, then what is X, Parhot’s regular annual payment?

(Spring 2011, test 1, question 9)

(Fall 2011, final, question 5)

(Spring 2012, final, question 3)

(Spring 2014, test 1, question 10)

(Spring 2016, test 1, question 8)

This is a problem where we need to find the payment amount associated with the present value of an annuity and an extra cash flow.

Timeline tip for FNAN 303: “everything” is annual so the timeline period is a year

The annual payment can not be found in one step on the financial calculator.

Approach

1) Find the present value of the extra payment made in 3 years

2) Find the present value of the stream of regular payments

3) Find the amount of each regular payment

1) Find the present value of the extra payment made in 3 years

The present value of a -$600 cash flow in 3 years at an annual rate of 9.6% is -600/1.0963 = -455.74

2) Find the present value of the stream of regular payments

The present value of all cash flows associated with all loan payments is -3,600

If the -600 cash flow in 3 years has a present value of -455.74, then the present value of the 7 annual fixed cash flows that start in 1 year and end in 7 years is -3,600 – (-455.74) = -3,144.26

3) Find the amount of each regular payment

Find the payment associated with an annuity with a present value of -3,144.26, a total of 7 payments, and a periodic discount rate of 9.6%

END mode

Enter 7 9.6 3,144.26 0

N I% PV PMT FV

Solve for -637.37

The regular annual payment would be $637.37

30. Crosby just took out a loan from the bank for $16,700. He plans to repay this loan by making a special payment to the bank of $5,300 in 5 quarters and by also making equal, regular quarterly payments of X. If the interest rate on the loan is 8.91 percent per quarter, he makes his first regular quarterly payment later today, and he makes his last regular quarterly payment in 6 quarters, then what is X, the amount of the regular quarterly payment?

(Fall 2016, test 1, question 6)

(Spring 2018, test 1, question 7)

This is a problem where we need to find the payment amount associated with the present value of an annuity due and an extra cash flow.

Timeline tip for FNAN 303: the CFs occur quarterly so the timeline period is a quarter

The quarterly payment can not be found in one step on the financial calculator.

Approach

1) Find the present value of the extra payment made in 5 quarters

2) Find the present value of the stream of regular payments

3) Find the amount of each regular payment

1) Find the present value of the extra payment made in 5 quarters

The present value of a -$5,300 cash flow in 5 quarters at a quarterly rate of 8.91% is

-5,300/1.08915 = -3,458.89

2) Find the present value of the stream of regular payments

The present value of all cash flows associated with all loan payments is -16,700

If the cash flow of -5,300 in 5 quarters has a present value of -3,458.89, then the present value of the 7 quarterly fixed cash flows that start immediately and end in 6 quarters is

-16,700 – (-3,458.89) = -13,241.11

3) Find the amount of each regular payment

Find the payment associated with an annuity due with a present value of -13,241.11, a total of 7 payments, and a periodic discount rate of 8.91%

BEGIN mode

Enter 7 8.91 13,241.11 0

N I% PV PMT FV

Solve for -2,408.36

The regular quarterly payment would be $2,408.36

31. An investment, which is worth $56,000 and has an expected return of 17.23 percent, is expected to pay fixed annual cash flows for a given amount of time. The first annual cash flow is expected in 1 year from today and the last annual cash flow is expected in 4 years from today. What is the present value of the annual cash flow that is expected in 3 years from today?

(Fall 2015, test 1, question 7)

(Fall 2017, test 1, question 7)

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 4-period annuity, so the cash flow expected in 3 years is the same as the cash flow expected each of the 4 times when cash flows take place

END mode

Enter 4 17.23 -56,000 0

N I% PV PMT FV

Solve for 20,506

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 = $20,506

r = .1723

PV0 = $20,506 / 1.17233

= $12,728

(Spring 2014, test 2, question 1)

(Fall 2014, final, question 4)

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 5-period annuity due, so the cash flow expected in 3 years is the same as the cash flow expected at each of the 5 times when cash flows take place

BEGIN mode

Enter 5 14.72 -62,000 0

N I% PV PMT FV

Solve for 16,016

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 = $16,016

r = .1472

PV0 = $16,016 / 1.14723

= $10,608

33. Jenny is buying a town house priced at $275,000. Mortgage A calls for her to make equal monthly payments for 15 years at a monthly interest rate of 0.80% with her first payment due in 1 month. However, her loan officer has offered her a new opportunity involving equal monthly payments for 20 years at a monthly interest rate of 0.75% with her first payment due later today. By how much would switching from mortgage A to the new opportunity reduce the amount of Jenny's monthly loan payment?

(Spring 2017, test 1, question 8)

Steps:

1. Find the payment with the mortgage A

2. Find the payment with the new opportunity

3. Find the payment with the mortgage A minus the payment with the new opportunity

Timeline tip for FNAN 303: the cash flows occur monthly so the timeline period is a month

1. Find the payment with the mortgage A

This is a problem where we need to find the payment amount associated with the present value of an annuity

N = 15 × 12 = 180 since monthly payments are made for 15 years

END mode since the first equal monthly payment is due in 1 month

END mode

Enter 180 0.80 275,000 0

N I% PV PMT FV

Solve for -2,888.23

Jenny’s monthly payment would be $2,888.23 with mortgage A

2. Find the payment with the new opportunity

This is a problem where we need to find the payment amount associated with the present value of an annuity due

N = 20 × 12 = 240 since monthly payments are made for 20 years

BEGIN mode since the first equal monthly payment is due later today

BEGIN mode

Enter 240 0.75 275,000 0

N I% PV PMT FV

Solve for -2,455.83

Jenny’s monthly payment would be $2,455.83 under the new opportunity

3. Find the payment with the mortgage A minus the payment with the new opportunity

Jenny’s monthly payment would be $2,888.23 under mortgage A

Jenny’s monthly payment would be $2,455.83 under the new opportunity

Switching from mortgage A to the new opportunity would reduce Jenny’s monthly payment by $2,888.23 – $2,455.83 = $432.40

34. Britta wants to buy a car that is available at two dealerships. The price of the car is the same at both dealerships. Best Buggies would let her make quarterly payments of $2,200 for 5 years at a quarterly interest rate of 3.80 percent. Her first payment to Best Buggies would be due in 3 months. If California Cars would let her make equal monthly payments for 4 years at a monthly interest rate of 1.30 percent and if her first payment to California Cars would be today, then how much would each monthly payment to California Cars be?

(Spring 2013, test 2, question 1)

The car price equals the opposite of the present value of the loan payments, discounted by the interest rate of the loan.

To solve:

1) Find the price of the car by discounting the loan payments to Best Buggies

2) Find the monthly payment needed for the present value of the loan payments to California Cars to equal the amount from 1 (in magnitude), the price of the car

1) Find the price of the car by discounting the loan payments to Best Buggies

Timeline tip for FNAN 303: the cash flows occur quarterly so the timeline period is a quarter

N = 5 years × 4 quarters per year = 20

I% = 3.80

PMT = -2,200

FV = 0

END mode since her first payment to Best Buggies would be due in 3 months

END mode

Enter 20 3.80 -2,200 0

N I% PV PMT FV

Solve for 30,435

The price of the car = $30,435

2) Find the monthly payment needed for the present value of the loan payments to California Cars to equal the amount from 1 (in magnitude), the price of the car

Timeline tip for FNAN 303: the cash flows occur monthly so the timeline period is a month

N = 4 years × 12 months per year = 48

I% = 1.30

PV = price of the car = $30,435

FV = 0

BEGIN mode since her first payment to California Cars would be due today

BEGIN mode

Enter 48 1.30 30,435 0

N I% PV PMT FV

Solve for -845.33

Britta’s monthly payment to California Cars would be $845.33

35. Camille just borrowed $59,000. She plans to repay this loan by making a special payment of $8,400 in 4 years and by making regular annual payments of $6,200 per year until the loan is paid off. If the interest rate on the loan is 9.70 percent per year and she makes her first regular annual payment of $6,200 immediately, then how many regular annual payments of $6,200 must Camille make?

(Fall 2011, test 2, question 3)

(Spring 2013, final, question 4)

(Fall 2013, final, question 5)

(Fall 2016, final, question 3)

(Spring 2018, final, question 3)

This is a problem where we need to find the number of payments associated with the present value of an annuity due. It is made a bit more complicated, because there is an extra payment.

Approach

1) Find the present value of the special payment

2) Find the present value of the regular payments

3) Find the number of regular payments

1) Find the present value of the special payment

The present value of a -$8,400 cash flow in 4 years at an annual rate of 9.70%

is -8,400/1.09704

= 5,800.33

2) Find the present value of the regular payments

We know that the present value of all payments = -$59,000

= present value of special payment + present value of regular payments

So -$59,000 = -$5,800.33 + present value of regular payments

So present value of regular payments = -$59,000 + $5,800.33 = -$53,199.67

3) Find the number of regular payments

Find the number of annuity due loan payments of $6,200 needed to have a present value of $53,199.67 (in magnitude) when the discount rate is 9.70%

BEGIN mode

Enter 9.70 53,199.67 -6,200 0

N I% PV PMT FV

Solve for 15.36

Camille would need to make 15.36 payments

36. Laina just borrowed $5,000. She plans to repay this loan by making a special payment of $2,000 in 2 years and by making regular annual payments of $800 per year until the loan is paid off. If the interest rate on the loan is 8.00 percent per year and she makes her first regular annual payment of $800 in 1 year, then how many regular annual payments of $800 must Laina make?

(Fall 2014, test 2, question 1)

(Spring 2015, final, question 5)

This is a problem where we need to find the number of payments associated with the present value of an annuity. It is made a bit more complicated, because there is an extra payment.

Approach

1) Find the present value of the special payment

2) Find the present value of the regular payments

3) Find the number of regular payments

1) Find the present value of the special payment

The present value of a -$2,000 cash flow in 2 years at an annual rate of 8.00%

is -2,000/1.08002 = 1,714.68

2) Find the present value of the regular payments

We know that the present value of all payments = -$5,000

= present value of special payment + present value of regular payments

So -$5,000 = -$1,714.68 + present value of regular payments

So present value of regular payments = -$5,000 + $1,714.68 = -$3,285.32

3) Find the number of regular payments

Find the number of annuity loan payments of $800 needed to have a present value of $3,285.32 (in magnitude) when the discount rate is 8.00%

END mode

Enter 8.00 3,285.32 -800 0

N I% PV PMT FV

Solve for 5.18

Laina would need to make 5.18 payments

37. Brenna wants to buy a car that is available at two dealerships. The price of the car is the same at both dealerships. Best Buggies would let her make quarterly payments of $2,250 for 5 years at a quarterly interest rate of 3.82 percent. Her first payment to Best Buggies would be due immediately. If California Cars would let her make equal monthly payments of $920 at a monthly interest rate of 1.35 percent and if her first payment to California Cars would be in 1 month, then how many monthly payments would Brenna need to make to California Cars?

(Spring 2018, test 1, question 8)

The car price equals the opposite of the present value of the loan payments, discounted by the interest rate of the loan.

To solve:

1) Find the price of the car by discounting the loan payments to Best Buggies

2) Find the number of payments needed for the present value of the loan payments to California Cars to equal the amount from step 1 (in magnitude), the price of the car

1) Find the price of the car by discounting the loan payments to Best Buggies

N = 5 years × 4 quarters per year = 20

I% = 3.82

PMT = -2,250

FV = 0

BEGIN mode since her first payment to Best Buggies would be due immediately

BEGIN mode

Enter 20 3.82 -2,250 0

N I% PV PMT FV

Solve for 32,258

The price of the car = $32,258

2) Find the number of payments needed for the present value of the loan payments to California Cars to equal the amount from step 1 (in magnitude), the price of the car

I% = 1.35

PV = price of the car = $32,258

PMT = -920

FV = 0

END mode since her first payment to California Cars would be due in 1 month

END mode

Enter 1.35 32,258 -920 0

N I% PV PMT FV

Solve for 47.82

Brenna would need to make 47.82 monthly payments to California Cars

38. Aldo wants to borrow $12,000 from the bank and is choosing among two possible loans. The interest rate on both loans is 1.4 percent per month. Loan A would require him to make 60 equal monthly payments, with the first payment made to the bank in 1 month. Loan B would also require him to make equal monthly payments to the bank. However, 1) the monthly payment associated with loan B would be $30 less than the monthly payment associated with loan A, and 2) the first monthly payment for loan B would be made to the bank later today. How many monthly payments to the bank must be made with loan B?

(Spring 2012, test 1, question 10)

This is a problem where we need to find the number of payments associated with the present value of an annuity due. However, before being able to answer that question, we need to find the payment amount associated with the present value of an annuity (loan A), in order to find the payment for loan B.

Timeline tip for FNAN 303: the cash flows occur monthly so the timeline period is a month

Approach:

1) Find the loan payment for loan A

2) Find the loan payment for loan B

3) Find the number of payments needed for loan B

1) Find the loan payment for loan A

END mode

Enter 60 1.4 12,000 0

N I% PV PMT FV

Solve for -296.94

Loan A requires equal monthly payments of $296.94

2) Find the loan payment for loan B

Loan B payment = loan A payment – $30.00 = $296.94 - $30.00 = $266.94

3) Find the number of payments needed for loan B

BEGIN mode

Enter 1.4 12,000 -266.94 0

N I% PV PMT FV

Solve for 69.72

Loan B requires 69.72 equal monthly payments of $266.94

Answers may differ slightly due to rounding

39. A factory is worth $3,200. It is expected to produce equal monthly cash flows of $300 for 9 months with the first monthly cash flow expected later today. The factory is also expected to make an extra cash flow of $1,000 in 9 months. What is the monthly cost of capital for the factory?

This is a problem where we need to find the rate associated with the present value of an annuity due and an extra payment

Timeline tip for FNAN 303: the cash flows occur monthly so the timeline period is a month

The rate can be found in one step on the financial calculator.

The 9 cash flows associated with the monthly cash flows of 300 are captured by setting PMT to 300 and N to 9, and by putting the calculator in BEGIN mode. Since the extra cash flow of 1,000 is expected in 9 months and since N is set to 9, FV can be set to 1,000. PV can be set to -3,200 as that is the opposite of the value of the factory.

BEGIN mode

Enter 9 -3,200 300 1,000

N I% PV PMT FV

Solve for 2.82

The monthly expected return is 2.82%

40. A factory is worth $75,300. It is expected to produce equal monthly cash flows of $7,160 for 9 months with the first monthly cash flow expected in 1 month. The factory is also expected to make an extra cash flow of $14,500 in 9 months. What is the monthly cost of capital for the factory?

This is a problem where we need to find the rate associated with the present value of an annuity and an extra payment

Timeline tip for FNAN 303: the cash flows occur monthly so the timeline period is a month

The rate can be found in one step on the financial calculator.

The 9 cash flows associated with the monthly cash flows of 7,160 are captured by setting PMT to 7,160 and N to 9, and by putting the calculator in END mode. Since the extra cash flow of 14,500 is expected in 9 months and since N is set to 9, FV can be set to 14,500. PV can be set to -75,300 as that is the opposite of the value of the factory.

END mode

Enter 9 -75,300 7,160 14,500

N I% PV PMT FV

Solve for 0.83

The monthly expected return is 0.83%

41. Bryna wants to buy a car that is available at two dealerships. The price of the car is the same at both dealerships. Best Buggies would let her make quarterly payments of $2,240 for 5 years at a quarterly interest rate of 3.72 percent. Her first payment to Best Buggies would be due in 3 months. If California Cars would let her make equal monthly payments of $935 for 4 years and if her first payment to California Cars would be today, then what is the monthly interest rate that Bryna would be charged by California Cars?

The car price equals the opposite of the present value of the loan payments, discounted by the interest rate of the loan.

To solve:

1) Find the price of the car by discounting the loan payments to Best Buggies

2) Find the discount rate needed for the present value of the loan payments to California Cars to equal the amount from step 1 (in magnitude), the price of the car

1) Find the price of the car by discounting the loan payments to Best Buggies

N = 5 years × 4 quarters per year = 20

I% = 3.72

PMT = -2,240

FV = 0

END mode since her first payment to Best Buggies would be due in 3 months

END mode

Enter 20 3.72 -2,240 0

N I% PV PMT FV

Solve for 31,211

The price of the car = $31,211

2) Find the discount rate needed for the present value of the loan payments to California Cars to equal the amount from step 1 (in magnitude), the price of the car

N = 4 years × 12 months per year = 48

PV = price of the car = $31,211

PMT = -935

FV = 0

BEGIN mode since her first payment to California Cars would be due today

BEGIN mode

Enter 48 31,211 -935 0

N I% PV PMT FV

Solve for 1.67

Bryna would be charged a monthly interest rate of 1.67% by California Cars

(Fall 2017, test 1, question 8)

The car price equals the opposite of the present value of the loan payments, discounted by the interest rate of the loan.

To solve:

1) Find the price of the car by discounting the loan payments to Best Buggies

2) Find the discount rate needed for the present value of the loan payments to California Cars to equal the amount from step 1 (in magnitude), the price of the car

1) Find the price of the car by discounting the loan payments to Best Buggies

N = 5 years × 4 quarters per year = 20

I% = 3.82

PMT = -2,125

FV = 0

BEGIN mode since her first payment to Best Buggies would be due immediately

BEGIN mode

Enter 20 3.82 -2,125 0

N I% PV PMT FV

Solve for 30,466

The price of the car = $30,466

2) Find the discount rate needed for the present value of the loan payments to California Cars to equal the amount from step 1 (in magnitude), the price of the car

N = 4 years × 12 months per year = 48

PV = price of the car = $30,466

PMT = -885

FV = 0

END mode since her first payment to California Cars would be due in 1 month

END mode

Enter 48 30,466 -885 0

N I% PV PMT FV

Solve for 1.45

Bella would be charged a monthly interest rate of 1.45% by California Cars

43. Bertina plans to retire in 4 years. She plans to collect annual payments of $53,800 for several years. Her first annual payment is expected to be received in 5 years and her last annual payment is expected in 11 years. She can earn 10.23 percent per year. How much money does Bertina expect to have in 4 years?

Even though the problem asks how much Bertina expects to have in the future, this is a problem where we need to find the present value of an annuity. A timeline with “re-timing” illustrates this when the “new time 0” is 4 years from today.

N = 7 since there are 7 payments

END mode since the first payment takes place 1 period after the reference point, which is at retirement in 4 years from today

END mode

Enter 7 10.23 53,800 0

N I% PV PMT FV

Solve for -259,949

Bertina expects to have $259,949 in 4 years from today

44. Martina plans to retire in 5 years. She plans to collect annual payments of $49,100 for several years. Her first annual payment is expected to be received in 5 years and her last annual payment is expected in 11 years. She can earn 12.99 percent per year. How much money does Martina expect to have in 5 years?

(Fall 2016, test 1, question 7)

Even though the problem asks how much Martina expects to have in the future, this is a problem where we need to find the present value of an annuity due. A timeline with “re-timing” illustrates this when the “new time 0” is 5 years from today.

N = 7 since there are 7 payments

BEGIN mode since the first payment takes place at the reference point, which is at retirement in 5 years from today

BEGIN mode

Enter 7 12.99 49,100 0

N I% PV PMT FV

Solve for -245,434

Martina expects to have $245,434 in 5 years from today

45. Padma plans to retire in 3 years with $317,000 in her account, which has an annual return of 4.36 percent. If she receives annual payments of X, with her first payment of X received in 3 years and her last payment of X received in 9 years, then what is X, the amount of each payment?

This is a problem where we need to find the payment associated with the present value of an annuity due.

Timeline tip for FNAN 303: “everything” is annual, so the timeline period is a year

The present value of the annuity due is the amount in her retirement account at retirement = $317,000

The periodic interest rate is 4.36 percent

There are no other cash flows or values to take into account

BEGIN Mode

Enter 7 4.36 -317,000 0

N I% PV PMT FV

Solve for 51,284

Padma will receive $51,284 per year

It is important to recognize that this is a problem where we want to find the payment associated with the present value of an annuity due. The only issue is that the annuity due starts cash flows in 3 years. However, we know the present value as of 3 years from now, so we can just re-set the timeline and set time 3 as the “new” time 0. It is essentially the same problem as if Padma has $317,000 in her account today, was retiring today, and was expecting the first of 7 payments today.

46. Gwen plans to retire in 3 years with $426,000 in her account, which has an annual return of 6.29 percent. If she receives annual payments of X, with her first payment of X received in 4 years and her last payment of X received in 9 years, then what is X, the amount of each payment?

(Spring 2015, test 1, question 10)

This is a problem where we need to find the payment associated with the present value of an annuity.

Timeline tip for FNAN 303: “everything” is annual, so the timeline period is a year

The present value of the annuity is the amount in her retirement account at retirement = $426,000

The periodic interest rate is 6.29 percent

There are no other cash flows or values to take into account

END Mode

Enter 6 6.29 -426,000 0

N I% PV PMT FV

Solve for 87,423

Gwen will receive $87,423 per year

It is important to recognize that this is a problem where we want to find the payment associated with the present value of an annuity. The only issue is that the annuity starts cash flows in 4 years. However, we know the present value as of 3 years from now, so we can just re-set the timeline and set time 3 as the “new” time 0. It is essentially the same problem as if Gwen has $426,000 in her account today, was retiring today, and was expecting the first of 6 payments in 1 year.

47. Emerson plans to retire in 3 years with $296,000 in his account, which has an annual return of 10.13 percent. If he receives payments of $60,700 per year and he receives his first $60,700 payment in 4 years, then how many payments of $60,700 can Emerson expect to receive?

(Fall 2015, test 1, question 8)

This is a problem where we need to find the number of payments associated with the present value of an annuity

END mode

Enter 10.13 -296,000 60,700 0

N I% PV PMT FV

Solve for 7.06

Emerson can expect to receive 7.06 payments

It is important to recognize that this is a problem where we want to find the number of payments associated with the present value of an annuity. The only issue is that the annuity starts cash flows in 4 years. However, we know the present value as of 3 years from now, so we can just re-set the timeline and set time 3 as the “new” time 0. It is essentially the same problem as if Emerson has $296,000 in his account today, was retiring today, and was expecting the first payment of $60,700 in 1 year.

48. Jozy plans to retire in 3 years with $358,000 in his account, which has an annual return of 9.72 percent. If he receives payments of $57,200 per year and he receives his first $57,200 payment in 3 years, then how many payments of $57,200 can Jozy expect to receive?

(Spring 2016, test 1, question 9)

This is a problem where we need to find the number of payments associated with the present value of an annuity due

BEGIN mode

Enter 9.72 -358,000 57,200 0

N I% PV PMT FV

Solve for 8.72

Jozy can expect to receive 8.72 payments

It is important to recognize that this is a problem where we want to find the number of payments associated with the present value of an annuity due. The only issue is that the annuity due starts cash flows in 3 years. However, we know the present value as of 3 years from now, so we can just re-set the timeline and set time 3 as the “new” time 0. It is essentially the same problem as if Jozy has $358,000 in his account today, was retiring today, and was expecting the first payment of $57,200 today.

49. Trace plans to retire in 3 years with $518,000 in his account. If he receives payments of $123,400 per year and he receives his first $123,400 payment in 3 years and his last $123,400 payment in 7 years, then what is the expected annual return for his account?

(Spring 2018, test 1, question 9)

This is a problem where we need to find the expected return associated with the present value of an annuity due

BEGIN mode

Enter 5 -518,000 123,400 0

N I% PV PMT FV

Solve for 9.60

Trace’s account has an expected annual return of 9.60 percent

It is important to recognize that this is a problem where we want to find the expected return associated with the present value of an annuity due. The only issue is that the annuity due starts cash flows in 3 years. However, we know the present value as of 3 years from now, so we can just re-set the timeline and set time 3 as the “new” time 0. It is essentially the same problem as if Trace has $518,000 in his account today, was retiring today, and was expecting the first payment of $123,400 later today.

50. Stone plans to retire in 3 years with $491,000 in his account. If he receives payments of $97,900 per year and he receives his first $97,900 payment in 4 years and his last $97,900 payment in 9 years, then what is the expected annual return for his account?

(Spring 2017, test 1, question 9)

This is a problem where we need to find the expected return associated with the present value of an annuity

END mode

Enter 6 -491,000 97,900 0

N I% PV PMT FV

Solve for 5.38

Stone’s account has an expected annual return of 5.38 percent

It is important to recognize that this is a problem where we want to find the expected return associated with the present value of an annuity. The only issue is that the annuity starts cash flows in 3 years. However, we know the present value as of 3 years from now, so we can just re-set the timeline and set time 3 as the “new” time 0. It is essentially the same problem as if Stone has $491,000 in his account today, was retiring today, and was expecting the first payment of $97,900 in one year from today.

51. What is the value of a building that is expected to generate no cash flows for several years and then generate fixed annual cash flows of $97,100 per year forever if the first annual $97,100 cash flow is expected in 5 years and the appropriate discount rate for the building is 10.2 percent?

(Fall 2010, test 2, question 1)

(Fall 2009, test 2, question 4)

(Spring 2012, test 2, question 3)

(Spring 2014, test 2, question 2)

(Fall 2014, final, question 5)

(Spring 2016, test 1, question 10)

(Fall 2016, test 1, question 8)

(Spring 2018, test 1, question 10)

The value can be found in 2 steps:

1) Find the value of the building as of one year before the first payment is made

2) Find the present value (as of today) of the building

1) The first of an infinite stream of constant payments occurs in 5 years, so we can find the value of the building as of 4 years from now as a fixed perpetuity, since 4 years from now is one year before 5 years from now, when the regular, fixed payments start

PV4 = C / r

C = $97,100

r = .102

PV4 = 97,100 / .102 = $951,961

In 4 years, the building is expected to have a value of $951,961

2) The value today of something expected to have a value of $951,961 in 4 years can be found as

PV0 = PVt / (1+r)t

t = 4

PVt = PV4 = 951,961

r = .102

PV0 = PV4 / (1+r)4 = 951,961 / (1.102)4 = 645,495

Note: it’s equivalent to express this as PV0 = PVk / (1+r)k = PV4 / (1+r)4 = 951,961 / (1.102)4 = 645,495

Mode is not relevant, since PMT = 0

Enter 4 10.2 0 951,961

N I% PV PMT FV

Solve for -645,495

Today, the value of the building is $645,495

(Spring 2010, test 2, question 4)

(Spring 2011, test 1, question 10)

(Fall 2012, test 2, question 3)

(Fall 2013, test 2, question 3)

(Fall 2014, test 2, question 2)

(Spring 2015, test 1, question 11)

(Fall 2015, test 1, question 9)

(Fall 2017, test 1, question 9)

The value can be found in 2 steps:

1) Find the value of the building as of one year before the first payment is made

2) Find the present value (as of today) of the building

1) The first of an infinite stream of constantly-growing payments occurs in 4 years, so we can find the value of the building as of 3 years from now as a growing perpetuity, since 3 years from now is one year before 4 years from now, when the regular, constantly-growing payments start

PV3 = C4 / (r – g)

C4 = $22,300

r = .061

g = .014

PV3 = 22,300 / (.061 – .014) = 22,300 / (.047) = 474,468

In 3 years, the building is expected to have a value of $474,468

2) The value today of something expected to have a value of $474,468 in 3 years can be found as

PV0 = PVt / (1+r)t

t = 3

PVt = PV3 = 474,468

r = .061

PV0 = PV3 / (1+r)3 = 474,468 / (1.061)3 = 397,247

Note: it’s equivalent to express this as PV0 = PVk / (1+r)k = PV3 / (1+r)3 = 474,468 / (1.061)3 = 397,247

Mode is not relevant, since PMT = 0

Enter 3 6.1 0 474,468

N I% PV PMT FV

Solve for -397,247

Today, the value of the building is $397,247

(Fall 2009, final, question 4)

(Spring 2011, test 2, question 2)

(Fall 2011, test 2, question 4)

(Spring 2013, test 2, question 2)

(Spring 2017, test 1, question 10)

The value can be found in 2 steps.

1) Find the value of the building as of one year before the first payment is made

2) Find the present value (as of today) of the building

1) The first of a series of 7 fixed payments occurs in 7 years, so we can find the value of the building as of 6 years from now as an ordinary annuity. The 7 fixed payments occur in 7, 8, 9, 10, 11, 12, and 13 years from now.

END Mode

Enter 7 9.6 78,900 0

N I% PV PMT FV

Solve for -389,230

2) The value today of something worth 389,230 in 6 years can be found as

PV0 = PV6 / (1+r)6

PV6 = 389,230

r = .096

PV0 = 389,230 / (1.096)6 = 224,565

Note: it’s equivalent to express this as PV0 = PVk / (1+r)k = PV6 / (1+r)6 = 389,230 / (1.096)6 = 224,565

Mode is not relevant, since PMT = 0

Enter 6 9.6 0 389,230

N I% PV PMT FV

Solve for -224,565

The value of the building is $224,565

36