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The assignment content presents a scenario where an investor is considering purchasing a 20-year bond with a par value of $8,500 and a coupon rate of 3.2%. The market interest rate has risen to 7.5%, causing the bond to sell at a discount. To understand interest rate risk, we must consider how bond values change when market rates fluctuate. Longer-term bonds are more sensitive to changes in market interest rates than shorter-term bonds. Using TVM templates, we can see that when the market rate increases from 10% to 15%, the value of the 25-year bond decreases significantly more than the 1-year bond. This shows how longer-term bonds are more affected by interest rate changes.

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Corporate Finance

Phantom Student

1

Phantom Student

1

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Table of Contents

Title Page 1

Table of Contents 2

Grading Rubric 3

Question 1 and Answer 6

Question 2 and Answer 10

Question 3 and Answer 13

Question 4 and Answer 14

Question 5 and Answer 16

References 18

2

Title Page 1

Table of Contents 2

Grading Rubric 3

Question 1 and Answer 6

Question 2 and Answer 10

Question 3 and Answer 13

Question 4 and Answer 14

Question 5 and Answer 16

References 18

2

Grading Rubric

Requirements Deductions

Quantitative Understanding 1

Qualitative Understanding 2

Teachability 2

Instructions 0

Total Score 95

Instructor Comments (if any):

A few errors, but otherwise excellent work.

3

Requirements Deductions

Quantitative Understanding 1

Qualitative Understanding 2

Teachability 2

Instructions 0

Total Score 95

Instructor Comments (if any):

A few errors, but otherwise excellent work.

3

FIN

Corporate Finance Assigment

Instructions: Read these instructions and each question carefully. Failure to follow these

instructions will result in deductions from your grade. This is a take home assignment, and it is

open note and open book. You may use internet resources. Additional instructions follow:

On are before 11:59pm of Wednesday the 28rd of September, 2016, each student will

electronically submit their assignment in MS Word via email to *************. The

student will title the email “Last Name, First Name – assignment”. The student will title

the Microsoft document “Last Name, First Name – Midterm”. The student will create a

title page and put their name and date on this title page.

If there is a conflict between these instructions and any prior information provided,

including your syllabus, these instructions take precedence.

If you believe a question is unanswerable, explain why and make any necessary

assumption(s) and answer the question. Make these assumptions at the beginning of your

answer.

If you believe a question has incorrect information, you are to make explicit reference to

the incorrect information and then restate the question with your corrected information

and answer the question. Make these corrections at the beginning of your answer.

If you leave a question unanswered, partial credit cannot be granted. Therefore, make an

attempt to answer all questions. Show your work for the maximum award of partial

credit. In general, show your work. It helps the grader decipher your thought process.

As you have significant time to complete the assignment, your work product is to be

organized and presented at a professional level of work, i.e. do not submit “sloppy” work.

See sample exams in Old assignment and Answers folder/link in the course shell.

Include a Title page, Table of Contents, and the Grading Rubric (found below) in that

order as your first three pages of your assignment.

Assume the reader of your assignment has no clear understanding of finance. However,

he/she is sound mathematically and has good comprehension and reasoning skills.

Use the attached TVM template to exhibit financial calculations where applicable. Do

not resubmit the blank template with your assignment.

If needed, attach a Reference section to the end of your assignment. You may follow any

citation and referencing style that you desire; however, you must be consistent with its

use.

Where applicable use the same notation as in class.

When writing out equations, use the Equation Editor, which can be found under the Insert

Tab in Microsoft Word.

Throughout the assignment you are being asked to replicate topics/learning materials.

You MAY NOT use examples from class, the textbook, other textbooks, the internet, or

any other outside sources. You must always create your own work product. Do not

plagiarize.

Assignments are to be submitted in a Microsoft Word document or be readable in

Microsoft Word. The readability of the electronic file/document that contains your

assignment is your responsibility.

The goal of your assignment should be to teach the subject matter.

Teach me. Teach yourself. Teach the world.

4

Corporate Finance Assigment

Instructions: Read these instructions and each question carefully. Failure to follow these

instructions will result in deductions from your grade. This is a take home assignment, and it is

open note and open book. You may use internet resources. Additional instructions follow:

On are before 11:59pm of Wednesday the 28rd of September, 2016, each student will

electronically submit their assignment in MS Word via email to *************. The

student will title the email “Last Name, First Name – assignment”. The student will title

the Microsoft document “Last Name, First Name – Midterm”. The student will create a

title page and put their name and date on this title page.

If there is a conflict between these instructions and any prior information provided,

including your syllabus, these instructions take precedence.

If you believe a question is unanswerable, explain why and make any necessary

assumption(s) and answer the question. Make these assumptions at the beginning of your

answer.

If you believe a question has incorrect information, you are to make explicit reference to

the incorrect information and then restate the question with your corrected information

and answer the question. Make these corrections at the beginning of your answer.

If you leave a question unanswered, partial credit cannot be granted. Therefore, make an

attempt to answer all questions. Show your work for the maximum award of partial

credit. In general, show your work. It helps the grader decipher your thought process.

As you have significant time to complete the assignment, your work product is to be

organized and presented at a professional level of work, i.e. do not submit “sloppy” work.

See sample exams in Old assignment and Answers folder/link in the course shell.

Include a Title page, Table of Contents, and the Grading Rubric (found below) in that

order as your first three pages of your assignment.

Assume the reader of your assignment has no clear understanding of finance. However,

he/she is sound mathematically and has good comprehension and reasoning skills.

Use the attached TVM template to exhibit financial calculations where applicable. Do

not resubmit the blank template with your assignment.

If needed, attach a Reference section to the end of your assignment. You may follow any

citation and referencing style that you desire; however, you must be consistent with its

use.

Where applicable use the same notation as in class.

When writing out equations, use the Equation Editor, which can be found under the Insert

Tab in Microsoft Word.

Throughout the assignment you are being asked to replicate topics/learning materials.

You MAY NOT use examples from class, the textbook, other textbooks, the internet, or

any other outside sources. You must always create your own work product. Do not

plagiarize.

Assignments are to be submitted in a Microsoft Word document or be readable in

Microsoft Word. The readability of the electronic file/document that contains your

assignment is your responsibility.

The goal of your assignment should be to teach the subject matter.

Teach me. Teach yourself. Teach the world.

4

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Have fun and good luck.

5

5

Question 1: Assume that a public corporation has 3,000,000 shares outstanding. First, you are to

create the necessary Balance Sheets and Income Statements and then calculate the annual Cash

Flow from Assets (aka: CFFA or Free Cash Flows (FCF)) for this corporation. A constraint here

is that your CFFA must range between $6,000,000 and $10,000,000 annually. Second, after

calculating CFFA, you are to assume that this corporation is a no-growth perpetuity and estimate

its present value (aka: intrinsic value). Assume the market determined risk adjusted required rate

of return is 6.125% for this corporation. Said another way, you are to replicate and explain the

relevant parts of the textbook, notes, and lectures associated with CFFA and intrinsic value of the

firm estimations. Teach me the concepts.

Answer 1: Part 1: Calculating annual Cash Flow from Assets

Notice, that as this question is written, it is unanswerable. Specifically, no marginal tax rate (τ)

is provided making it impossible to estimate Net Income (and other necessary calculations).

Accordingly, let the marginal tax rate equal 55% for this firm.

JLB Design is a (fictional) publicly traded corporation in the fashion sector. Exhibit 1.1 and 1.2

are the company’s financial statements for 2012 and 2013.

Exhibit 1.1 Balance Sheet for JLB Design (2012/2013)

JLB Design

2012 and 2013 B/S

Assets Liabilities and Owners’ Equity

2012 2013 2012 2013

Current Assets Current liabilities

Cash $7,000,000 $6,000,000 A/P $25,000,0000 $26,000,000

A/R $27,000,000 $36,000,000 Notes Payable $9,000,000 $6,000,000

Inventory $45,000,000 $47,000,000 Total $34,000,000 $32,000,000

Total $79,000,000 $89,000,000

Fixed Assets Long term debt $31,000,000 $33,000,000

Plant and

Equipment

$64,000,000 $77,000,000 Owner’s Equity

Common stock and

paid-in surplus

$26,000,000 $36,000,000

Retained earnings $52,000,000 $65,000,000

Total $78,000,000 $101,000,000

Total assets $143,000,000 $166,000,000 Total liabilities and owner’s

equity

$143,000,000 $166,000,000

Exhibit 1.2 Income Statement for JLB Design (2013)

JLB Design

2013 Income Statement

Net sales $156,000,000

Cost of goods sold (COGs) $80,000,000

Depreciation $2,000,000

Earnings before interest and taxes (EBIT) $74,000,000

Interest paid $2,500,000

Taxable income $71,500,000

Taxes (55%) $39,325,000

6

create the necessary Balance Sheets and Income Statements and then calculate the annual Cash

Flow from Assets (aka: CFFA or Free Cash Flows (FCF)) for this corporation. A constraint here

is that your CFFA must range between $6,000,000 and $10,000,000 annually. Second, after

calculating CFFA, you are to assume that this corporation is a no-growth perpetuity and estimate

its present value (aka: intrinsic value). Assume the market determined risk adjusted required rate

of return is 6.125% for this corporation. Said another way, you are to replicate and explain the

relevant parts of the textbook, notes, and lectures associated with CFFA and intrinsic value of the

firm estimations. Teach me the concepts.

Answer 1: Part 1: Calculating annual Cash Flow from Assets

Notice, that as this question is written, it is unanswerable. Specifically, no marginal tax rate (τ)

is provided making it impossible to estimate Net Income (and other necessary calculations).

Accordingly, let the marginal tax rate equal 55% for this firm.

JLB Design is a (fictional) publicly traded corporation in the fashion sector. Exhibit 1.1 and 1.2

are the company’s financial statements for 2012 and 2013.

Exhibit 1.1 Balance Sheet for JLB Design (2012/2013)

JLB Design

2012 and 2013 B/S

Assets Liabilities and Owners’ Equity

2012 2013 2012 2013

Current Assets Current liabilities

Cash $7,000,000 $6,000,000 A/P $25,000,0000 $26,000,000

A/R $27,000,000 $36,000,000 Notes Payable $9,000,000 $6,000,000

Inventory $45,000,000 $47,000,000 Total $34,000,000 $32,000,000

Total $79,000,000 $89,000,000

Fixed Assets Long term debt $31,000,000 $33,000,000

Plant and

Equipment

$64,000,000 $77,000,000 Owner’s Equity

Common stock and

paid-in surplus

$26,000,000 $36,000,000

Retained earnings $52,000,000 $65,000,000

Total $78,000,000 $101,000,000

Total assets $143,000,000 $166,000,000 Total liabilities and owner’s

equity

$143,000,000 $166,000,000

Exhibit 1.2 Income Statement for JLB Design (2013)

JLB Design

2013 Income Statement

Net sales $156,000,000

Cost of goods sold (COGs) $80,000,000

Depreciation $2,000,000

Earnings before interest and taxes (EBIT) $74,000,000

Interest paid $2,500,000

Taxable income $71,500,000

Taxes (55%) $39,325,000

6

Net income $32,175,000

Dividends $19,175,000

Addition to retained earnings $13,000,000

The balance sheet provides a snapshot of a company’s financial situation by breaking down the

company’s assets, liabilities and equity at a specific point in time. The income statement

summarizes a company’s financial performance over a specific period of time. Using the

information provided in these financial statements, it is possible to calculate the annual Cash

Flow from Assets (CFFA) or Free Cash Flows (FCF) of a company.

CFFA are defined as “the total of cash flow to creditors and cash flow to stockholders, consisting

of the following: operating cash flow, capital spending and change in net working capital.”

Therefore it can be broken down two ways:

1. Cash flow identity: taking the sum of total cash flows to creditors and the total cash flows

to stockholders (the entities that supply capital to the company)

2. Or, by taking the operating cash flow less capital spending and changes in net working

capital during a given year

Using information in the balance sheet and income statements from above, we can calculate the

CFFA for JLB Design for 2013 using the 2nd method once you calculate the three components

below:

1. Operating Cash Flow

2. Capital Spending

3. Changes in New Working Capital (NWC)

1. Operating Cash Flow: Operating cash flow is defined as a company’s cash inflow that

resulted from production and sales. Exhibit 1.3 shows the steps to calculate JLB Design’s

operating cash flow. We can find the values needed for this calculation in the income

statement. Of note, it is important to remember to adjust for depreciation because it was not

a cash deduction but we will need to deduct taxes since they are considered a cash flow. In

this case, we would not include interest in our calculations because it is considered a

financing expense.

Exhibit 1.3 Operating Cash Flow for JLB Design (2013)

JLB Design

2013 Operating Cash Flow

EBIT $74,000,000

+Depreciation +$2,000,0000

-taxes (τ = 55%) -$39,325,000

Operating cash flows $36,675,000

2. Capital Spending: Capital spending is defined as the amount spent on fixed assets less any

money that was earned from the sale of the fixed assets. For this calculation shown in Exhibit

1.4, we get the values for ending net fixed assets and beginning net fixed assets from the

7

Dividends $19,175,000

Addition to retained earnings $13,000,000

The balance sheet provides a snapshot of a company’s financial situation by breaking down the

company’s assets, liabilities and equity at a specific point in time. The income statement

summarizes a company’s financial performance over a specific period of time. Using the

information provided in these financial statements, it is possible to calculate the annual Cash

Flow from Assets (CFFA) or Free Cash Flows (FCF) of a company.

CFFA are defined as “the total of cash flow to creditors and cash flow to stockholders, consisting

of the following: operating cash flow, capital spending and change in net working capital.”

Therefore it can be broken down two ways:

1. Cash flow identity: taking the sum of total cash flows to creditors and the total cash flows

to stockholders (the entities that supply capital to the company)

2. Or, by taking the operating cash flow less capital spending and changes in net working

capital during a given year

Using information in the balance sheet and income statements from above, we can calculate the

CFFA for JLB Design for 2013 using the 2nd method once you calculate the three components

below:

1. Operating Cash Flow

2. Capital Spending

3. Changes in New Working Capital (NWC)

1. Operating Cash Flow: Operating cash flow is defined as a company’s cash inflow that

resulted from production and sales. Exhibit 1.3 shows the steps to calculate JLB Design’s

operating cash flow. We can find the values needed for this calculation in the income

statement. Of note, it is important to remember to adjust for depreciation because it was not

a cash deduction but we will need to deduct taxes since they are considered a cash flow. In

this case, we would not include interest in our calculations because it is considered a

financing expense.

Exhibit 1.3 Operating Cash Flow for JLB Design (2013)

JLB Design

2013 Operating Cash Flow

EBIT $74,000,000

+Depreciation +$2,000,0000

-taxes (τ = 55%) -$39,325,000

Operating cash flows $36,675,000

2. Capital Spending: Capital spending is defined as the amount spent on fixed assets less any

money that was earned from the sale of the fixed assets. For this calculation shown in Exhibit

1.4, we get the values for ending net fixed assets and beginning net fixed assets from the

7

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balance sheet and once again adjust for depreciation since it was deducted in the company’s

income statement.

Exhibit 1.4 Capital Spending for JLB Design (2013)

JLB Design

Capital Spending 2013

Ending Net fixed assets 2013 $77,000,000

-beginning net fixed asset 2013 $64,000,000

+ depreciation $2,000,000

NET Capital spending $15,000,000

3. Changes in net working capital: Changes in the net working capital (NWC) is defined as the

difference between the net working capital at the end and beginning of a year. In our

example, we can subtract the beginning NWC of 2013 (also the same as the ending NWC of

2012) from the ending NWC of 2013. NWC is calculated by subtracting total current

liabilities from total current assets (values from the balance sheet). The calculations below

are used to arrive at the NWC values used to calculate changes in NWC in Exhibit 1.5.

NWC2013=Total Current Asset2013-Total Current Liabilities2013

NWC2013=$89,000,000 - $32,000,000

NWC2013=$57,000,000

NWC2012=Total Current Assets2012-Total Current Liabilities2012

NWC2012=$79,000,000 - 34,000,000

NWC2012=$45,000,000

Exhibit 1.5 Changes in Net Working Capital for JLB Design (2013)

JLB Design

Changes in Net Working Capital 2013

Ending Net working capital (NWC) 2013 $57,000,000

-Beginning NWC 2013 45,000,000

Change in NWC $12,000,000

Finally, once the operating cash flow, net capital spending and changes in NWC for 2013 have

been calculated, the CFFA can be calculated as shown in Exhibit 1.6.

Exhibit 1.6 Calculating Cash Flow for JLB Design (2013)

JLB Design

2013 CFFA

Operating Cash Flow $36,675,000

-Net capital spending $15,000,000

-changes in NWC $12,000,000

CFFA2013 $9,675,000

In order to check this CFFA calculation, we can use the first method of cash flow identity to

ensure the calculations were correct. Since we know that CFFA is the total cash flow to creditors

and equity holders, we can use the information from our financial statements to calculate CFFA

as shown in Exhibit 1.7.

8

income statement.

Exhibit 1.4 Capital Spending for JLB Design (2013)

JLB Design

Capital Spending 2013

Ending Net fixed assets 2013 $77,000,000

-beginning net fixed asset 2013 $64,000,000

+ depreciation $2,000,000

NET Capital spending $15,000,000

3. Changes in net working capital: Changes in the net working capital (NWC) is defined as the

difference between the net working capital at the end and beginning of a year. In our

example, we can subtract the beginning NWC of 2013 (also the same as the ending NWC of

2012) from the ending NWC of 2013. NWC is calculated by subtracting total current

liabilities from total current assets (values from the balance sheet). The calculations below

are used to arrive at the NWC values used to calculate changes in NWC in Exhibit 1.5.

NWC2013=Total Current Asset2013-Total Current Liabilities2013

NWC2013=$89,000,000 - $32,000,000

NWC2013=$57,000,000

NWC2012=Total Current Assets2012-Total Current Liabilities2012

NWC2012=$79,000,000 - 34,000,000

NWC2012=$45,000,000

Exhibit 1.5 Changes in Net Working Capital for JLB Design (2013)

JLB Design

Changes in Net Working Capital 2013

Ending Net working capital (NWC) 2013 $57,000,000

-Beginning NWC 2013 45,000,000

Change in NWC $12,000,000

Finally, once the operating cash flow, net capital spending and changes in NWC for 2013 have

been calculated, the CFFA can be calculated as shown in Exhibit 1.6.

Exhibit 1.6 Calculating Cash Flow for JLB Design (2013)

JLB Design

2013 CFFA

Operating Cash Flow $36,675,000

-Net capital spending $15,000,000

-changes in NWC $12,000,000

CFFA2013 $9,675,000

In order to check this CFFA calculation, we can use the first method of cash flow identity to

ensure the calculations were correct. Since we know that CFFA is the total cash flow to creditors

and equity holders, we can use the information from our financial statements to calculate CFFA

as shown in Exhibit 1.7.

8

Exhibit 1.7 Cash Flow Identity

JLB Design

2013 Cash flow to Creditors

Interest paid

Long term debt 2013 $33,000,000

-Long term debt 2012 $31,000,000

Net Borrowing 2013 $2,000,000

$2,500,000

-$2,000,000

Cash Flow to Creditors $500,000

JLB Design

2013 Cash flow to Equity Holders

Dividends paid

Common equity 2013 $36,000,000

-Common equity 2012 $26,000,000

Net Equity Raised 2013 $10,000,000

$19,175,000

-$10,000,000

Cash Flow to Stockholders $9,175,000

CFFA = Cash flow to creditors + Cash flow to stockholders

CFFA2013 = $500,000 + $9,175,000 = $9,675,000

Part 2: Intrinsic Value of the firm

Assuming that JLB Designs will “live forever” and its future cash flows will remain constant, we

can refer to JLB Designs as a “no growth perpetuity.” In this case, the given market-determined,

risk-adjusted required rate of return is 6.125%. This is in essences the company’s weighted

average cost of capital (WACC). Therefore, an estimated intrinsic value can be calculated as

follows:

ValueJLB= CFFA

( 1+WACC )1 + CFFA

( 1+ WACC )2 + CFFA

( 1+WACC )3 +… +CFFA

( 1+WACC )∞

Knowing thatPV JLB=∑

t =1

T CFFA

(1+WACC )t will converge to PV JLB= CFFA

WACC as t approaches ∞.

We can therefore estimate the value of JLB Designs:

PV JLB= CFFA

WACC = $ 9,675,000

0.06125 = $157,959,184 (rounded to nearest dollar)

Question 2: In class, we discussed the formulas for the present value of a perpetuity (aka: no-

growth perpetuity) and the present value of an ordinary annuity (aka: finite series of cash flows):

PV = CF

r (I)

PV =∑

i=1

n CF

(1+r )i (II)

9

JLB Design

2013 Cash flow to Creditors

Interest paid

Long term debt 2013 $33,000,000

-Long term debt 2012 $31,000,000

Net Borrowing 2013 $2,000,000

$2,500,000

-$2,000,000

Cash Flow to Creditors $500,000

JLB Design

2013 Cash flow to Equity Holders

Dividends paid

Common equity 2013 $36,000,000

-Common equity 2012 $26,000,000

Net Equity Raised 2013 $10,000,000

$19,175,000

-$10,000,000

Cash Flow to Stockholders $9,175,000

CFFA = Cash flow to creditors + Cash flow to stockholders

CFFA2013 = $500,000 + $9,175,000 = $9,675,000

Part 2: Intrinsic Value of the firm

Assuming that JLB Designs will “live forever” and its future cash flows will remain constant, we

can refer to JLB Designs as a “no growth perpetuity.” In this case, the given market-determined,

risk-adjusted required rate of return is 6.125%. This is in essences the company’s weighted

average cost of capital (WACC). Therefore, an estimated intrinsic value can be calculated as

follows:

ValueJLB= CFFA

( 1+WACC )1 + CFFA

( 1+ WACC )2 + CFFA

( 1+WACC )3 +… +CFFA

( 1+WACC )∞

Knowing thatPV JLB=∑

t =1

T CFFA

(1+WACC )t will converge to PV JLB= CFFA

WACC as t approaches ∞.

We can therefore estimate the value of JLB Designs:

PV JLB= CFFA

WACC = $ 9,675,000

0.06125 = $157,959,184 (rounded to nearest dollar)

Question 2: In class, we discussed the formulas for the present value of a perpetuity (aka: no-

growth perpetuity) and the present value of an ordinary annuity (aka: finite series of cash flows):

PV = CF

r (I)

PV =∑

i=1

n CF

(1+r )i (II)

9

With CF> 0 and 0< r< 1. We know that PV = CF

r converges to PV =∑

i=1

n CF

(1+r )i as n approaches

∞. Create a numerical example to show that this is so and explain the relevance of this

convergence with respect to the estimation of financial asset valuation.

Answer 2:

There is an incorrect statement here. Actually, the present value of an annuity (PVannuity)

converges to the present value of a perpetuity (PVperpetuity) as n approaches ∞. This relationship

illustrates how you can estimate the value of a long-lifed asset that does not have growth in

payments as long as you have a good estimate for payment/cash flows (CF) and a good estimate

of the discount rate (r).

More specifically:

In order to understand this relationship it is important to recognize some key terms. An

(ordinary) annuity have cash flows that arrive at the end of a period. Present value of an

annuity (PVAT or PVAn) is the current value of the cash flows that would arrive at the end of the

period, given a specific discount rate (r).

For example, if I want to know the current value (aka: Present Value) of getting $1,400 a year for

five years at a discount rate of 5%, I could use the present value of an annuity formula:

Exhibit 2.2 Calculations:

PV = PMT

(1+r )1 + PMT

(1+ r)2 + PMT

( 1+r )3 + PMT

(1+r )4 + PMT

(1+r )5

PV = 1400

(1+.05)1 + 1400

(1+0.05)2 + 1400

( 1+ 0.05 ) 3 + 1400

(1+0.05)4 + 1400

(1+0.05)5

PV =$ 6061.27

Alternatively, I can use the Time Value of Money (TVM) template below (Exhibit 2.3) to

illustrate the same scenario and calculate the PV using Microsoft Excel or a financial calculator.

10

Exhibit 2.1 Equations

PV annuity=∑

t =1

T PMT

(1+r )t as n approaches ∞

→

PV perpetuity = PMT

r

PMT=payment or cash flow

r=discount rate

T=total number of payments

r converges to PV =∑

i=1

n CF

(1+r )i as n approaches

∞. Create a numerical example to show that this is so and explain the relevance of this

convergence with respect to the estimation of financial asset valuation.

Answer 2:

There is an incorrect statement here. Actually, the present value of an annuity (PVannuity)

converges to the present value of a perpetuity (PVperpetuity) as n approaches ∞. This relationship

illustrates how you can estimate the value of a long-lifed asset that does not have growth in

payments as long as you have a good estimate for payment/cash flows (CF) and a good estimate

of the discount rate (r).

More specifically:

In order to understand this relationship it is important to recognize some key terms. An

(ordinary) annuity have cash flows that arrive at the end of a period. Present value of an

annuity (PVAT or PVAn) is the current value of the cash flows that would arrive at the end of the

period, given a specific discount rate (r).

For example, if I want to know the current value (aka: Present Value) of getting $1,400 a year for

five years at a discount rate of 5%, I could use the present value of an annuity formula:

Exhibit 2.2 Calculations:

PV = PMT

(1+r )1 + PMT

(1+ r)2 + PMT

( 1+r )3 + PMT

(1+r )4 + PMT

(1+r )5

PV = 1400

(1+.05)1 + 1400

(1+0.05)2 + 1400

( 1+ 0.05 ) 3 + 1400

(1+0.05)4 + 1400

(1+0.05)5

PV =$ 6061.27

Alternatively, I can use the Time Value of Money (TVM) template below (Exhibit 2.3) to

illustrate the same scenario and calculate the PV using Microsoft Excel or a financial calculator.

10

Exhibit 2.1 Equations

PV annuity=∑

t =1

T PMT

(1+r )t as n approaches ∞

→

PV perpetuity = PMT

r

PMT=payment or cash flow

r=discount rate

T=total number of payments

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Exhibit 2.3 Time Value of Money Template

INPUTS 5 periods 5% $1,400 $0

N I/YR PV PMT FV

OUTPUTS -$6,061.27

P/Y = 1

I/YR=Interest per Period

PV=Present Value

PMT=Payments

FV=Future Value

N=Number of Periods

P/Y=Payments per Period

In contrast to an ordinary annuity which is a finite series of cash flows, a perpetuity is a series of

cash flows that will last forever, where T = ∞. The present value of a perpetuity (PVperpetuity) is

calculated using the formula: PV perpetuity= PMT

r where the payment is constant and there is no

growth. Therefore the present value of $1,400 payments at a discount rate of 5% is $28,000.

PV = $ 1,400

.05 =$ 28,000

To illustrate that the PVannuity converges to the PVperpetuity as n approaches ∞, we can increase the

number of periods while holding the other variables/parameters constant:

For example, let N increase to 10 periods:

PV with 10 periods

INPUTS 10 5% $1400 0

N I/YR PV PMT FV

OUTPUTS -$10,810.43

P/Y = 1

Thus, doubling the number of periods does not double the present value. This should not surprise us in an

environment with positive discount rates.

Now, let N increase another 10-fold to N = 100 periods:

PV with 100 periods

INPUTS 100 5% $1400 0

N I/YR PV PMT FV

OUTPUTS -$27,787.07

P/Y = 1

11

INPUTS 5 periods 5% $1,400 $0

N I/YR PV PMT FV

OUTPUTS -$6,061.27

P/Y = 1

I/YR=Interest per Period

PV=Present Value

PMT=Payments

FV=Future Value

N=Number of Periods

P/Y=Payments per Period

In contrast to an ordinary annuity which is a finite series of cash flows, a perpetuity is a series of

cash flows that will last forever, where T = ∞. The present value of a perpetuity (PVperpetuity) is

calculated using the formula: PV perpetuity= PMT

r where the payment is constant and there is no

growth. Therefore the present value of $1,400 payments at a discount rate of 5% is $28,000.

PV = $ 1,400

.05 =$ 28,000

To illustrate that the PVannuity converges to the PVperpetuity as n approaches ∞, we can increase the

number of periods while holding the other variables/parameters constant:

For example, let N increase to 10 periods:

PV with 10 periods

INPUTS 10 5% $1400 0

N I/YR PV PMT FV

OUTPUTS -$10,810.43

P/Y = 1

Thus, doubling the number of periods does not double the present value. This should not surprise us in an

environment with positive discount rates.

Now, let N increase another 10-fold to N = 100 periods:

PV with 100 periods

INPUTS 100 5% $1400 0

N I/YR PV PMT FV

OUTPUTS -$27,787.07

P/Y = 1

11

Notice that present value is closing in on $28,000 and we are only 100 periods forward, which is far

closer to today than to an infinite time horizon.

Now, let N = 150 periods:

PV with 150 periods

INPUTS 150 5% $1400 0

N I/YR PV PMT FV

OUTPUTS -$27,981.43

P/Y = 1

When you continue to increase n towards ∞, the PVannuity will continue to converge to the

PVperpetuity ($28,000) but will it will never reach exactly $28,000. The mathematical explanation

for this is that, if all other variables remain the same, the denominator will increase exponentially

in size resulting in any $1,400 cash flow being worth only a fractional cent in present value

terms. Thus, future expected cash flows are worth virtually nothing in present values term. This

means that the vast majority of the present value of a perpetuity occurs within the early periods

of existence. This is quite convenient for business valuation purposes because we can quickly

and rather accurately estimate the market value of a finite series of cash flows that have

relatively long expected lives by way of the present value of a perpetuity, requiring only accurate

estimates of both CF (aka: PMT) and r.

12

closer to today than to an infinite time horizon.

Now, let N = 150 periods:

PV with 150 periods

INPUTS 150 5% $1400 0

N I/YR PV PMT FV

OUTPUTS -$27,981.43

P/Y = 1

When you continue to increase n towards ∞, the PVannuity will continue to converge to the

PVperpetuity ($28,000) but will it will never reach exactly $28,000. The mathematical explanation

for this is that, if all other variables remain the same, the denominator will increase exponentially

in size resulting in any $1,400 cash flow being worth only a fractional cent in present value

terms. Thus, future expected cash flows are worth virtually nothing in present values term. This

means that the vast majority of the present value of a perpetuity occurs within the early periods

of existence. This is quite convenient for business valuation purposes because we can quickly

and rather accurately estimate the market value of a finite series of cash flows that have

relatively long expected lives by way of the present value of a perpetuity, requiring only accurate

estimates of both CF (aka: PMT) and r.

12

Question 3: Consider PV =∑

i=1

n CF

(1+r )i which expands to PV = CF

(1+r )1 + CF

(1+ r)2 +…+ CF

(1+r )n

In terms of present value (PV), what happens to the right hand side of the equation as r

approaches zero (0)?

Answer 3:

When calculating present value (PV), r represents the discount rate. As r approaches zero, the

denominator of the present value formula approaches one. Therefore, as r approaches zero, the

sum of the future expected cash flows get continually closer to PV.

Expressed in equation form, if PV =∑

i=1

n CF

(1+r )i and r =0, thenPV =∑

i=1

n

CF.

13

i=1

n CF

(1+r )i which expands to PV = CF

(1+r )1 + CF

(1+ r)2 +…+ CF

(1+r )n

In terms of present value (PV), what happens to the right hand side of the equation as r

approaches zero (0)?

Answer 3:

When calculating present value (PV), r represents the discount rate. As r approaches zero, the

denominator of the present value formula approaches one. Therefore, as r approaches zero, the

sum of the future expected cash flows get continually closer to PV.

Expressed in equation form, if PV =∑

i=1

n CF

(1+r )i and r =0, thenPV =∑

i=1

n

CF.

13

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Question 4: Considering interest changes and bond prices, develop an example to explain how a

bond selling at a premium arises and how a bond selling at a discount arises. Use the relevant

parts of the textbook, notes, and lectures to teach me the concept.

Answer 4:

First, for the example of a bond selling at a premium, we will use a 7 year, semiannually

compounded bond with a par value of $5,000 and a coupon rate of 8% per period or 16%

annually. At issuance, assuming the market rate equals the coupon rate, which is most common.

V B =$ 400 [ 1

.08 − 1

.08 ( 1+.08 )14 ]+ $ 5,000

( 1+.08 )14

This can be shown by way of a Time Value of Money template as well:

INPUTS 14 periods 8% $400 $5,000

N I/P PV PMT FV

OUTPUT $5,000

P/Y = 1

Now suppose we have moved forward 3 years and the market interest rate has fallen to 5% per

period ( rd =10 % annually).

V B =$ 400 [ 1

.05 − 1

.05 ( 1+.05 ) 8 ] + $ 5,000

( 1+ .05 ) 8

Again, using a TVM template:

INPUTS 8 periods 5% $400 $5,000

N I/P PV PMT FV

OUTPUT $5,969.48

P/Y = 1

This bond is said to be selling at a premium. It is offering an interest payment of 8% per period

of its par value. New bonds, issued at this point in time, will only pay 5% per period of their par

value. Therefore, investors are willing to pay a premium to acquire this asset because it is

offering a rate of return that is higher than the market is currently offering (8% per period vs 5%

per period) on comparable bonds.

Second, let’s look at an example of a bond selling at a discount. For the discount bond example,

we will use a 20 year, annually compounded bond with a par value of $8,500 and a coupon rate

of 3.2%. At issuance, assuming the market rate equals the coupon rate, which is most common:

14

bond selling at a premium arises and how a bond selling at a discount arises. Use the relevant

parts of the textbook, notes, and lectures to teach me the concept.

Answer 4:

First, for the example of a bond selling at a premium, we will use a 7 year, semiannually

compounded bond with a par value of $5,000 and a coupon rate of 8% per period or 16%

annually. At issuance, assuming the market rate equals the coupon rate, which is most common.

V B =$ 400 [ 1

.08 − 1

.08 ( 1+.08 )14 ]+ $ 5,000

( 1+.08 )14

This can be shown by way of a Time Value of Money template as well:

INPUTS 14 periods 8% $400 $5,000

N I/P PV PMT FV

OUTPUT $5,000

P/Y = 1

Now suppose we have moved forward 3 years and the market interest rate has fallen to 5% per

period ( rd =10 % annually).

V B =$ 400 [ 1

.05 − 1

.05 ( 1+.05 ) 8 ] + $ 5,000

( 1+ .05 ) 8

Again, using a TVM template:

INPUTS 8 periods 5% $400 $5,000

N I/P PV PMT FV

OUTPUT $5,969.48

P/Y = 1

This bond is said to be selling at a premium. It is offering an interest payment of 8% per period

of its par value. New bonds, issued at this point in time, will only pay 5% per period of their par

value. Therefore, investors are willing to pay a premium to acquire this asset because it is

offering a rate of return that is higher than the market is currently offering (8% per period vs 5%

per period) on comparable bonds.

Second, let’s look at an example of a bond selling at a discount. For the discount bond example,

we will use a 20 year, annually compounded bond with a par value of $8,500 and a coupon rate

of 3.2%. At issuance, assuming the market rate equals the coupon rate, which is most common:

14

V B =$ 272 [ 1

.032 − 1

.032 ( 1+.032 )20 ]+ $ 8,500

( 1+.032 )20

Again, using a TVM template:

INPUTS 20 periods 3.2% $272 $8,500

N I/P PV PMT FV

OUTPUT $8,500

P/Y = 1

Now suppose we have moved forward 14 years, and the market interest rate ( rd ) has risen to

7.5%. Now,

V B =$ 272 [ 1

.075 − 1

.075 ( 1+.075 ) 6 ] + $ 8,500

( 1+.075 ) 6

INPUTS 6 periods 7.5% $272 $8,500

N I/P PV PMT FV

OUTPUT $6,784.40

P/Y = 1

This bond is said to be selling at a discount. Investors can purchase newly issued bonds of

comparable risk paying interest of 7.5% of their par value. Why would an investor want to only

earn 3.2%? In order to sell this bond under the current market conditions, the bond holder must

offer a discount and sell the bond at less than its face value in order to move the security. Said

another way, this bond with its 3.2% coupon is priced to yield 7.5% at $6,784.40.

15

.032 − 1

.032 ( 1+.032 )20 ]+ $ 8,500

( 1+.032 )20

Again, using a TVM template:

INPUTS 20 periods 3.2% $272 $8,500

N I/P PV PMT FV

OUTPUT $8,500

P/Y = 1

Now suppose we have moved forward 14 years, and the market interest rate ( rd ) has risen to

7.5%. Now,

V B =$ 272 [ 1

.075 − 1

.075 ( 1+.075 ) 6 ] + $ 8,500

( 1+.075 ) 6

INPUTS 6 periods 7.5% $272 $8,500

N I/P PV PMT FV

OUTPUT $6,784.40

P/Y = 1

This bond is said to be selling at a discount. Investors can purchase newly issued bonds of

comparable risk paying interest of 7.5% of their par value. Why would an investor want to only

earn 3.2%? In order to sell this bond under the current market conditions, the bond holder must

offer a discount and sell the bond at less than its face value in order to move the security. Said

another way, this bond with its 3.2% coupon is priced to yield 7.5% at $6,784.40.

15

Question 5: Develop an explain interest rate risk. Basically, you are to replicate and

explain the relevant parts of your textbook, notes, and lectures. Make sure that you

highlight the fact that shorter-term bonds have more interest rate sensitivity than longer-term

bonds. Teach me the concept

There is an incorrect statement here. Specifically, shorter-term bonds are not more sensitive to

interest rate risk than longer-term bonds. In fact, it is longer-term bonds that suffer more than

shorter-term bonds to interest rate risk exposure.

Interest rate risk is the risk of bond value changes associated with fluctuations in the market-

determined, risk-adjusted required rate of return (rd). Bonds with varying maturity lengths

have different sensitivities to fluctuations in the market interest rate. Shorter-term bonds, or

bonds closer to maturity, are less sensitive to interest rate changes and as a result they are said to

suffer less interest rate sensitivity. Conversely, longer-term bonds have a greater interest rate

risks and are greatly affected by changes in the market rate.

Suppose you have two 10% coupon bonds, one with a maturity of 1 year and the other with a

maturity of 25 years. All else equal, the 1-year bond will be less sensitive to changes in the

market rate than the 25-year bond. Each bond is annually compounded and its par value is

$1,500. Exhibit 5.1a and 5.1b illustrate this phenomenon graphically and plots the changes that

occur to bond value when the market rate is changed from 10% to 5% and 15% and 20%. At a

10% market rate, rd=rc and par value of the bond is $1500 as expected. The steepness of the

slopes of the graphs show how the shorter-term bond does not change in value as much as the

longer-term bond when the market rate changes.

Exhibit 5.2 shows the TVM templates used to calculate the bond value for the 1-year bond in

comparison to a 25-year bond.

Exhibit 5.1a Interest Rate Risk and Time to Maturity

Interest Rate 1 year bond 25 year bond

5% $1,571.43 $2,557.05

10% $1,500.00 $1,500.00

15% $1,434.78 $1,015.19

20% $1,375.00 $757.86

16

explain the relevant parts of your textbook, notes, and lectures. Make sure that you

highlight the fact that shorter-term bonds have more interest rate sensitivity than longer-term

bonds. Teach me the concept

There is an incorrect statement here. Specifically, shorter-term bonds are not more sensitive to

interest rate risk than longer-term bonds. In fact, it is longer-term bonds that suffer more than

shorter-term bonds to interest rate risk exposure.

Interest rate risk is the risk of bond value changes associated with fluctuations in the market-

determined, risk-adjusted required rate of return (rd). Bonds with varying maturity lengths

have different sensitivities to fluctuations in the market interest rate. Shorter-term bonds, or

bonds closer to maturity, are less sensitive to interest rate changes and as a result they are said to

suffer less interest rate sensitivity. Conversely, longer-term bonds have a greater interest rate

risks and are greatly affected by changes in the market rate.

Suppose you have two 10% coupon bonds, one with a maturity of 1 year and the other with a

maturity of 25 years. All else equal, the 1-year bond will be less sensitive to changes in the

market rate than the 25-year bond. Each bond is annually compounded and its par value is

$1,500. Exhibit 5.1a and 5.1b illustrate this phenomenon graphically and plots the changes that

occur to bond value when the market rate is changed from 10% to 5% and 15% and 20%. At a

10% market rate, rd=rc and par value of the bond is $1500 as expected. The steepness of the

slopes of the graphs show how the shorter-term bond does not change in value as much as the

longer-term bond when the market rate changes.

Exhibit 5.2 shows the TVM templates used to calculate the bond value for the 1-year bond in

comparison to a 25-year bond.

Exhibit 5.1a Interest Rate Risk and Time to Maturity

Interest Rate 1 year bond 25 year bond

5% $1,571.43 $2,557.05

10% $1,500.00 $1,500.00

15% $1,434.78 $1,015.19

20% $1,375.00 $757.86

16

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0% 5% 10% 15% 20%

$0.00

$500.00

$1,000.00

$1,500.00

$2,000.00

$2,500.00

$3,000.00

$2,557.05

$1,500.00

$1,015.19 $757.86

$1,571.43 $1,434.78 $1,375.00

Exhibit 5.1b Interest Rate Risk and Time to Maturity

1 year bond 25 year bond

The TVM Templates below indicate the calculations used to calculate the bond values with a

10% coupon rate with fluctuating interest rates for a 1-year bond versus a 25-year bond.

Exhibit 5.2: TVM Templates used to illustrate Interest Rate Risk

5% interest, 1-year bond

INPUTS 1 5% +$150 $1500

N I/YR PV PMT FV

OUTPUTS -$1,571.43

1/Y=1

10% interest, 1-year bond

INPUTS 1 10% +$150 $1500

N I/YR PV PMT FV

OUTPUTS -$1500.00

1/Y=1

15% interest, 1-year bond

INPUTS 1 15% +$150 $1500

N I/YR PV PMT FV

OUTPUTS -$1,434.78

1/Y=1

20% interest, 1-year bond

INPUTS 1 20% +$150 $1500

N I/YR PV PMT FV

OUTPUTS -$1,375.00

1/Y=1

17

$0.00

$500.00

$1,000.00

$1,500.00

$2,000.00

$2,500.00

$3,000.00

$2,557.05

$1,500.00

$1,015.19 $757.86

$1,571.43 $1,434.78 $1,375.00

Exhibit 5.1b Interest Rate Risk and Time to Maturity

1 year bond 25 year bond

The TVM Templates below indicate the calculations used to calculate the bond values with a

10% coupon rate with fluctuating interest rates for a 1-year bond versus a 25-year bond.

Exhibit 5.2: TVM Templates used to illustrate Interest Rate Risk

5% interest, 1-year bond

INPUTS 1 5% +$150 $1500

N I/YR PV PMT FV

OUTPUTS -$1,571.43

1/Y=1

10% interest, 1-year bond

INPUTS 1 10% +$150 $1500

N I/YR PV PMT FV

OUTPUTS -$1500.00

1/Y=1

15% interest, 1-year bond

INPUTS 1 15% +$150 $1500

N I/YR PV PMT FV

OUTPUTS -$1,434.78

1/Y=1

20% interest, 1-year bond

INPUTS 1 20% +$150 $1500

N I/YR PV PMT FV

OUTPUTS -$1,375.00

1/Y=1

17

5% interest, 25-year bond

INPUTS 25 5% +$150 $1500

N I/YR PV PMT FV

OUTPUTS -$2,557.05

1/Y=1

10% interest, 25-year bond

INPUTS 25 10% +$150 $1500

N I/YR PV PMT FV

OUTPUTS -$1500.00

1/Y=1

15% interest, 25-year bond

INPUTS 25 15% +$150 $1500

N I/YR PV PMT FV

OUTPUTS -$1,015.19

1/Y=1

20% interest, 25-year bond

INPUTS 25 20% +$150 $1500

N I/YR PV PMT FV

OUTPUTS -$757.86

1/Y=1

References:

Johnson, Ken (2015). Mini Lectures and Class Notes.

Ross, S., Westerfield, R., Jordan, B (2012). Fundamentals of Corporate Finance, 10th Edition.

18

INPUTS 25 5% +$150 $1500

N I/YR PV PMT FV

OUTPUTS -$2,557.05

1/Y=1

10% interest, 25-year bond

INPUTS 25 10% +$150 $1500

N I/YR PV PMT FV

OUTPUTS -$1500.00

1/Y=1

15% interest, 25-year bond

INPUTS 25 15% +$150 $1500

N I/YR PV PMT FV

OUTPUTS -$1,015.19

1/Y=1

20% interest, 25-year bond

INPUTS 25 20% +$150 $1500

N I/YR PV PMT FV

OUTPUTS -$757.86

1/Y=1

References:

Johnson, Ken (2015). Mini Lectures and Class Notes.

Ross, S., Westerfield, R., Jordan, B (2012). Fundamentals of Corporate Finance, 10th Edition.

18

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