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>Mini Case 1 0

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5 Chapter 4. Mini Case Situation Assume that you are nearing graduation and have applied for a job with a local bank. As part of the bank’s evaluation process, you have been asked to take an examination that covers several financial analysis techniques. The first section of the test addresses discounted cash flow analysis. See how you would do by answering the following questions.

a. Draw time lines for

(1) a $

100

lump sum cash flow at the end of

Year

2, (2) an ordinary annuity of $

100per year for

3years, and (3) an uneven cash flow stream of -$50, $100, $

75, and $50 at the end of

Years0 through 3.

FUTURE VALUE $100 lump sum at the end of year 2. I% Time period

0 1

2FV

at year end Or

dinary annuity of $100 per year for three years.

I%

Time period 0 1 2

FV at year end

I%

Time period 0 1 2 3FV at year end

b. (1.) What’s the future value of an initial $100 after 3 years if it is invested in an account paying 1 0%annual interest

? Interest rate
0.1
These are the basic inputs, in blue.
Cash flow
100

Time period 0 1 2 3

FV at year end

ote: This problem was solved using the formula, FVn

= PV(1+I)N. However, there are a number of ways the problem could have used Excel’s “Wizard Function”. First, you must select the Function wizard icon found in the toolbar at the top of the screen, which looks like this: fx. When you get the “Insert Function ” dialog box, select the category for Financial Functions, as shown below. After selecting the category for Financial functions, scroll down until you can selet the FV function, as show below. Alternatively, select the menu Formulas, then then select Financial, then pick FV. After selecting the “FV” function from the “Financial” category, we will be using the following dialog box to input our data. Notice that we entered a value instead of a cell reference as the input for the problem for instructional purposes. It’s really better to enter cell values so that your spreadsheet can automatically reflect any changes to the input data. This is one of the features that makes the spreadsheet such a valuable tool. <-- ALWAYS enter addresses, not numbers. Using the function wizard yields the following result: FV = Future Value Interest Factors With a spreadsheet, calculating FVIF’s is a simple operation, and we can use it to graph the relationship between future value, growth, interest rates, and time. A similar table can be found in the textbook, along with a corresponding graph. Period (N)

0%

5% 10% 15%
0

2

46 8

10

Relationships among Future Value, Growth, Interest Rates, and Time b. (2) What is the present value of $100 to be received in 3 years if the appropriate interest rate is 10%? PRESENT VALUE (PV) Simply put, the present value (PV) is the value today of some future cash flow (or series of cash flows). The interest rate used to “discount” a given cash flow is the opportunity cost rate, and is equivalent to the next best investment alternative of the same risk. PROBLEM Interest rate 10%

Cash flow 100

Number of Years Discounted BackTime period 0 1 2 3

PV 100.00 This problem can also be solved using the function wizard using a procedure similar to that for the FV function. Begin by putting the pointer on the cell in which you want to display the result. Then, after selecting the “PV” function from the “Paste Function” box, the input data for the problem must be entered. Then click OK to get the result, $75. 13. PV = c. We sometimes need to find how long it will take a sum of money (or anything else) to grow to some specified amount. For example, if a company’s sales are growing at a rate of

20% per year, how long will it take sales to double?
Finding Time to Double
I =
0.2
<-- Rule of 72: Divide 72 by the interest rate as a percentage to find doubling time.
Time period 0 1 2 ?

Present Value
$

0 2.00 Finding N, the number of periods Use the function NPER, as shown below. SOLVING FOR I

PROBLEM

d. If you want an investment to double in three years, what interest rate must it earn? N 3PV -1 FV 2

Once again, Excel has a special function for this calculation. We suggest using either a financial calculator or the function wizard to solve this type of problem, because of its complexity. The procedure can be carried out using the function wizard, by selecting the “Rate” function from the list of financial functions in the “Paste Function” dialog box. Upon entering the time, present value, and future value, the interest rate can be found. Note that you can either type the data in or else activate the menu slot and then click on the appropriate cell. I = We noted above the difficulty of solving this problem mathematically. This is because it involves taking the Nth root of a value (an operation which generally requires either a calculator or a computer). However, if you would like to know how to solve the problem mathematically, the formula is (FVn/PV)(1/N) – 1, which is derived from the FV formula.

N 3

PV 1 I =FV 2

e. What is the difference between an ordinary annuity and an annuity due? What type of annuity is shown below? How would you change it to the other type of annuity? See Ch 28Mini Case Show.ppt f. (1.) What is the future value of a 3-year ordinary annuity of $100 if the appropriate interest rate is 10%? FUTURE VALUE OF A

NANNUITY As explained below, one way to solve this problem is to find the future value of each of the annuity payments. However, this is tedious, especially if a lot of years are involved. In the following example, we use the input data of the interest rate and time to calculate the future value in time period 3 of each individual cash flow. Lastly, we take the sum of all the future values, which gives us the future value of the entire annuity.

N 3

I 0.1PMT

100

Time period 0 1 2 3

CFt Annuity’s FV: FV3 Σ= An easier procedure is to solving for the future value of an annuity with the function wizard. This procedure is similar to that of a lump sum future value. Whereas before we left the “Pmt” field blank, now we insert the annuity payment ($100 in this case). First, we access the “FV” function box from the list of financial functions. Then, we input our new data. A key thing to watch is the “Type” input box. Previously, we left this box alone. An “0” or no entry in the box indicates an ordinary annuity, and a “1” indicates an annuity due. Though we can leave the box blank, it is a good habit to enter a “0” in the field.FV =

PRESENT VALUE OF AN ANNUITY f. (2.) What is the present value of the annuity? N 3

I 0.1

PMT 100

Time period 0 1 2 3

=

Or, you could use the function wizard for this ordinary annuity.

PV =

f. (3.) What would the future and present values be if the annuity were an annuity due? The procedure for solving this problems follows the previous example with one notable exception. Since, the payments occur at the beginning of each year, the first annuity payment occurs in time period 0, and the last occurs in time period 2. N 3

I 0.1

PMT 100

Time period 0 1 2 3

Additionally, using the function wizard for this problem is exactly like above, but we enter a “1” instead of a “0” into the “Type” field.

FV =

To find the present value of the annuity due, this problem is solved just like the previous problem, except that the payments occur in periods 0 through 2. N 3

I 0.1

PMT 100

Time period 0 1 2 3

CFt Annuity PV

PV3 =

PV =

g. What is the present value of the following uneven cash flow stream? The appropriate interest rate is 10%, compounded annually. I = 10%Time period

0 1 2 3 4

0 100 30

0

300

-50 Cash Flows PV of Cash Flows 0

90.91 24

7.93 22

5.39 -34.15 NPV = = Σ of PVs = As we show above, the first way to solve for the present value of this uneven cash flow stream is to use the time line to find the present value of each of the cash flows in the periods in which they occur, then sum all the present values. This procedure will yield the correct present value. This problem could also be set up in a column format; it is a matter of personal preference as to which format is easier to interpret and use. Once we have converted our data into a data table, we can solve for the present value of each of the cash flows (like we did previously) and add all of the present values together.

I 0.1

N CFN PV0 0 01 100

2 300

3 300

4 -50

PV of CF

stream =
With, the financial calculator, we could enter each of these cash flows and the discount rate, and simply press NPV for the present value of the cash flow stream. In Excel, we can perform a similar calculation by using the “NPV” function. While this function is very similar, there is a key distinction. In the cash flow register of your calculator, the first entry you make would be the cash flow to occur in time period zero. However, the “NPV” function interprets the first data entry as being the cash flow in time period one. Therefore, the initial cash flow must be added seperately. In this particular example, the initial cash flow is zero.
Or

PV =

h. (1.) Identify (a) the stated, or quoted, or nominal rate (iNom) and (b) the periodic rate (iPER). Inputs INOM (quarterly)0.1

This is the rate stated in contracts. m=periods/yr2

This is the number of periods per year, m. The periodic is associated with the number of compounding periods per year. M = 4 quarterly, <td>

12for monthly, and

360 or 365 for annual compounding.
IPER =
inom/m
IPER = 10% / 2

IPER =

h. (2.) Will the future value be larger or smaller if we compound an initial amount more often than annually, for example, every 6 months (semiannually), holding the stated interest rate constant? Why?
Larger, because interest is earned on interest.
The effective annual rate is the annual rate that causes the PV to grow to the same FV as under multiple compounding periods.
EFF% =
(1+ INOM/M)M
EFF% = (1

1

EFF% =

SEMIANNUAL AND OTHER COMPOUNDING PERIODS
h. (3.) What is the future value of $100 after 5 years under 12% annual compounding?
N 3

I

FV =

PV 100

What is the FV with semiannual compounding?
N (years x 2)

6

I (I per year/2)
0.06

FV =

PV 100

What is the FV with quarterly compounding? N (years x 4) 12

I (I per year/4)
0.03

FV =

PV 100

What is the FV with monthly compounding? N (years x 12) 36

I (I per year/12)
0.01

FV =

PV 100

What is the FV with daily compounding? N (years x 365) 1095 I (I per year/12) 0.0003287671 FV =

PV 100

I. Will the effective annual rate ever be equal to the nominal (quoted) rate? Only if the compounding period is equal to 1 year. j. (1.) What would the required payment be on a $1,000 loan that is to be repaid in three equal installments at the end of each of the next three years if the interest rate is 10%? SETUP FOR A 30 Y EARMORTGAGE. GRAPH BELOW. THE LONGER THE MATURITY, THE SMALLER THE INITIAL PRINCIPAL PAYMENT. N 3

PMT = Total pmts Tot. int. paid Tot. prin. pdN 30 PMT =

$106.08 I 0.1 I 0.1PV 1000

PV 1000

Now, construct an amortization table for the loan described above.

N

Beg. Amt. PaymentInterest

Principal End. Amt. j. (2.) What is the annual interest expense for the borrower, and the annual interest income for the lender, during Year 2? 1

2

3

N Beg. Amt. Payment Interest Principal End. Amt. 4

1 5

2 6

3 7

8

9

10

Note: See Columns M
11
through R for a 30 year

12

mortgage example.

13

14
15

16
17
18
19
20

21
22

23
24

25
26
27
28

k. On January 1, you deposit $100 in an account that pays a nominal (or quoted) interest rate of 11.33463%, with interest added (compounded) daily. How much will you have in your account on October 1, or 9 months later? (

days)
29
30

$0.00

$0.00 $0.00

0 1 2 3 4 5 273

100

I 0.00031054Bart Kreps: 11.33463%/365 N 273

FV

l. (1.) What is the value at the end of Year 3 of the following cash flow stream if the quoted interest rate is 10%, compounded semiannually? Annual rate =

10%

Periods

per year = 2

Periodic rate = 5%

Years 0

1

1.52

2.5 3

Periods 0 1.0 2

4

5.0 6

Cash Flow 0 100 0 100 0 100

There are two approaches. First, you could simply find the future value of each cash flow using the period rate and compounded for the appropriate number of periods, as shown below.

Periods 0 1.0 2 3.0 4 5.0 6

FV of CF $121.55 $110.25 $100.00 Total FV = Σ = Alternatively, you could calculate the annual effective rate and use this to find the future value of a 3-year annuity. Annual effective rate = 10.25% FV = $331.80 l. (2.) What is the PV of the same stream? Using the first approach, we find the present value of each individual cash flow using the periodic rate and the number of periods. Periods 0 1 2 3.0 4 5.0 6PV of CF $90.70 $82.27 $74.62 Total FV = $247.59 In the second approach, we use the annual effective rate to find the present value of a 3-year annuity. PV =

$247.59

l. (3.) Is the stream an annuity? No, because we don’t have a payment for each compounding period.
l. (4.) An important rule is that you should never show a nominal rate on a time line or use it in calculations unless what condition holds? (Hint: Think of annual compounding, when iNOM = EAR = iPER.) What would be wrong with your answer to questions l(1) and l(2) if you used the nominal rate (10%) rather than the periodic rate (iNOM/2 = 10%/2 = 5%)? Use the nominal rate only for annual compounding.
m. Suppose someone offered to sell you a note calling for the payment of $1,000 in 15 months (or

days). They offer to sell it to you for $

850. You have $850 in a bank time deposit that pays a 6.76649% nominal rate with daily compounding, which is a

7% effective annual interest rate, and you plan to leave the money in the bank unless you buy the note. The note is not risky–you are sure it will be paid on schedule. Should you buy the note? Check the decision in three ways: (1) by comparing your future value if you buy the note versus leaving your money in the bank, (2) by comparing the PV of the note with your current bank account, and (3) by comparing the EFF% on the note versus that of the bank account.
See which provides the greater future wealth
0 1 2 3 4 5 456

850

I

Bank account:

FV < $1,000,

so buy the note. See which has the greater present value0 1 2 3 4 5 456

1000 I 0.00018538

N 456

PV

> $859 cost, so buy the note. See which has the higher effective rate of return, EFF%0 1 2 3 4 5 456

850 1000N 456

I per day EAR > 7% so buy the note.&P of &N

Relationships among Future Value, Growth, Interest Rate, and Time

0% interest rate 0 2 4 6 8 10 5% interest rate 0 2 4 6 8 10 10% interest rate 0 2 4 6 8 10 15% interest rate 0 2 4 6 8 10 Periods

Future Value of $1

Payment Distribution

Interest Principal

Year

Payment

Interest Principal

1

$100

2

$200

3

$200

5

$200

6

$0

4

$200

7

$1,000

>Problem

/

/1

Value of Money

5

of $1,

00 invested to earn

annually 5 years from now. Answer this question by using a math formula and also by using the E

cel function wizard.

00

5. Put the pointer on E12 and then click the function wizard (fx) to see the completed menu. Also, it is generally easiest to fill in the wizard menus by clicking on one of the menu slots to activate the cursor and then clicking on the cell where the item is given. Then, hit the tab key to move down to the next menu slot to continue filling out the dialog box.

, and

%

for 0, 1, 2, 3, 4, and 5 years. Then create a graph with years on the horizontal axis and FV on the vertical axis to display your results.

and the Column Input Cell is D10, and we set Cell B

equal to Cell E

. Then, we selected (highlighted) the range B32:E3

, then clicked Data, Table, and filled in the menu items to complete the table.

s (D10):

Rate (D9)

0% 5% 20%0

1

2

3

4

5

:E

. Then click the chart wizard. Then follow the menu. It is easy to make a chart, but a lot of detailed steps are involved to format it so that it’s “pretty.” Pretty charts are generally not necessary to get the picture, though. Note that as the last item in the chart menu you are asked if you want to put the chart on the worksheet or on a separate tab. This is a matter of taste. We put the chart below on the spreadsheet so we could see how changes in the data lead to changes in the graph.

%), then to .4, then to .5, etc., to see how the table and the chart changes.

Inputs:0

10%

Formula:

because there are no periodic payments. Also, set the FV with a negative sign so that the PV will appear as a positive number.

Inputs:0

I/YR = ?N = 5

per year. How long would it take for the population to double?

Inputs: PV = FV = 2%

%

. Then find the FV of that same annuity.

Inputs: N = 5 15%

PV =

FV =

x =

x =

1000 1000

N = 5 10

Formula:

Wizard (FV):

Orig. Inputs New Inputs

I/YR = 10% 5%

N = 5 10

Formula: PV = FV/(1+I)^N =

Wizard (PV):

.

Year 1 1002 200

3

0

8%

Year Payment x

= FV

.

2 2006.00

3 400PV =

,000. The interest rate is 8%, and you must amortize the loan over 10 years with equal end-of-year payments. Set up an amortization schedule that shows the annual payments and the amount of each payment that repays the principal and the amount that constitutes interest expense to the borrower and interest income to the lender.

10

1.

YearPmt Interest

1

2

3

4

5

7

8

9

10

4 and D1

, and change the interest rate and the term to maturity to see how the payments would change.

payments in all, with the same original amount and the same nominal interest rate. What would the amortization schedule show now?

66

%.

Beg. Amt. Pmt Interest Principal End. Bal.

1

2

3

4

5

6

7

8

9

10

12

15

16

17

18

20

21

30

32

33

38

40

45

47

50

60

6466

67

85

100

120Solution | 7 | 1 | 6 | 5 | |||||||||||||||

Chapter: | 4 | Time | |||||||||||||||||

Problem: | |||||||||||||||||||

3 | |||||||||||||||||||

a. Find the | FV | 0 | 1 | 0% | x | To get the dialog box, click on fx, then Financial, then FV, then OK. | |||||||||||||

Inputs: | PV | = | 10 | ||||||||||||||||

I/YR = | 10% | ||||||||||||||||||

N = | |||||||||||||||||||

Formula: | FV = PV(1+I)^ | N = | |||||||||||||||||

Wizard (FV): | |||||||||||||||||||

Note: When you use the wizard and fill in the menu items, the result is the formula you see on the formula line if you click on cell E | 12 | ||||||||||||||||||

Experiment by changing the input values to see how quickly the output values change. | |||||||||||||||||||

b. Now create a table that shows the FV at 0%, | 5% | 20 | |||||||||||||||||

Begin by typing in the row and column labels as shown below. We could fill in the table by inserting formulas in all the cells, but a better way is to use an Excel data table as described in the model for Chapter 4 (Bond Valuation). We used the data table procedure. Note that the Row Input Cell is D | 9 | 32 | 11 | 8 | |||||||||||||||

Year | Interest | ||||||||||||||||||

To create the graph, first select the range C | 33 | 38 | |||||||||||||||||

Note that the inputs to the data table, hence to the graph, are now in the row and column heads. Change the 20% in Cell E32 to .3 (or | 30 | ||||||||||||||||||

c. Find the PV of $1,000 due in 5 years if the discount rate is 10% per year. Again, work the problem with a formula and also by using the function wizard. | |||||||||||||||||||

FV = | 100 | ||||||||||||||||||

I/YR = | |||||||||||||||||||

PV = FV/(1+I)^N = | |||||||||||||||||||

Wizard (PV): | |||||||||||||||||||

Note: In the wizard’s menu, use zero for | Pmt | ||||||||||||||||||

d. A security has a cost of $1,000 and will return $2,000 after 5 years. What rate of return does the security provide | ? | ||||||||||||||||||

PV = | -1000 | ||||||||||||||||||

FV = | 200 | ||||||||||||||||||

Wizard (Rate): | |||||||||||||||||||

Note: Use zero for Pmt since there are no periodic payments. Note that the PV is given a negative sign because it is an outflow (cost to buy the security). Also, note that you must scroll down the menu to complete the inputs. | |||||||||||||||||||

e. Suppose California’s population is 30 million people, and its population is expected to grow by | 2% | ||||||||||||||||||

-30 | |||||||||||||||||||

60 | |||||||||||||||||||

I/YR = growth rate | |||||||||||||||||||

Wizard (NPER): | = Years to double. | ||||||||||||||||||

f. Find the PV of an ordinary annuity that pays $1,000 at the end of each of the next 5 years if the interest rate is | 15 | ||||||||||||||||||

PMT = | $ 1,000 | ||||||||||||||||||

I/YR = | |||||||||||||||||||

PV: Use function wizard (PV) | |||||||||||||||||||

FV: Use function wizard (FV) | |||||||||||||||||||

g. How would the PV and FV of the above annuity change if it were an annuity due rather than an ordinary annuity? | |||||||||||||||||||

For the PV, each payment would be received one period sooner, hence would be discounted back one less year. This would make the PV larger. We can find the PV of the annuity due by finding the PV of an ordinary annuity and then multiplying it by (1 + I). | |||||||||||||||||||

PV annuity due = | |||||||||||||||||||

Exactly the same adjustment is made to find the FV of the annuity due. | |||||||||||||||||||

FV annuity due = | |||||||||||||||||||

h. What would the FV and the PV for parts a and c be if the interest rate were 10% with semiannual compounding rather than 10% with annual compounding? | |||||||||||||||||||

Part a. FV with semiannual compounding: | Orig. Inputs | New Inputs | |||||||||||||||||

PV = | |||||||||||||||||||

FV = PV(1+I)^N = | |||||||||||||||||||

Part c. PV with semiannual compounding: | |||||||||||||||||||

i. Find the PV and FV of an investment that makes the following end-of-year payments. The interest rate is | 8% | ||||||||||||||||||

Payment | |||||||||||||||||||

40 | |||||||||||||||||||

Rate = | |||||||||||||||||||

To find the PV, use the NPV function: | |||||||||||||||||||

Excel does not have a function for the sum of the future values for a set of uneven payments. Therefore, we must find this FV by some other method. Probably the easiest procedure is to simply compound each payment, then sum them, as is done below. Note that since the payments are received at the end of each year, the first payment is compounded for 2 years, the second for 1 year, and the third for 0 years. | |||||||||||||||||||

(1 + I )^(N-t) | |||||||||||||||||||

1. | 17 | 1 | 16 | 64 | |||||||||||||||

1.08 | 21 | ||||||||||||||||||

1.00 | 400.00 | ||||||||||||||||||

Sum = | |||||||||||||||||||

An alternative procedure for finding the FV would be to find the PV of the series using the NPV function, then compound that amount, as is done below: | |||||||||||||||||||

FV of PV = | |||||||||||||||||||

j. Suppose you bought a house and took out a mortgage for $ | 50 | ||||||||||||||||||

Original amount of mortgage: | 50000 | ||||||||||||||||||

Term of mortgage: | |||||||||||||||||||

Interest rate: | 0.08 | ||||||||||||||||||

Annual payment (use PMT function): | -$7, | 45 | 47 | ||||||||||||||||

Beg. Amt. | Principal | End. Bal. | |||||||||||||||||

(1) Create a graph that shows how the payments are divided between interest and principal repayment over time. | |||||||||||||||||||

Go back to cells D | 18 | 85 | |||||||||||||||||

(2) Suppose the loan called for 10 years of monthly payments, | 120 | ||||||||||||||||||

Now we would have a 12 × 10 = 120-payment loan at a monthly rate of .08/12 = 0. | 66 | 67 | |||||||||||||||||

The monthly payment would be: | |||||||||||||||||||

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>Mini Case 1 0

/2

8/1

5 Chapter 5. Mini Case Situation Sam Strother and Shawna Tibbs are vice –presidents of Mutual of Seattle Insurance Company and co-direct

ors of the company’s pension fund management division. A major new client, the

Northwestern Municipal Alliance, has requested that Mutual of Seattle present an investment seminar to the mayors of the represented cities, and Strother and Tibbs, who will make the actual presentation, have asked you to help them by answering the following questions. Because the Boeing Company operates in one of the league’s cities, you are to work Boeing into the presentation. a. What are the key features of a bond? Answer: b. What are call provisions and sinking fund provisions? Do these provisions make bonds more or less risky? Call Provisions and Sinking Funds A call provision that allows the issuer to redeem the bond at a specified time before the maturity date. If interest rates fall, the issuer can refund the bonds and issue new bonds at a lower rate. Because of this, borrowers are willing to pay more and lenders require more on callable bonds. In a sinking fund provision, the issuer pays off the loan over its life rather than all at the maturity date. A sinking fund reduces the risk to the investor and shortens the maturity. This is not good for investors if rates fall after issuance. c. How is the value of any asset whose value is based on expected future cash flows determined? Answer: d. How is the value of a bond determined? What is the value of a

10-year,

$1,000par value bond with a 10 percent annual coupon if its required rate of return is 10 percent? Finding the “Fair Value” of a Bond First, we list the key features of the bond as “model inputs”: Years to

Mat:
10

Coupon rate

: 1

0% Annual Pmt: $ 100 Par value = FV:
$1,000

Going rate, r

d: 10% The easiest way to solve this problem is to use Excel’s PV function. Click fx, then financial, then PV. Then fill in the menu items as shown in our snapshot in the screen shown just below. Value of bond = Thus, this bond sells at its par value. That situation always exists if the going rate is equal to the coupon rate. The PV function can only be used if the payments are constant, but that is normally the case for bonds. e. (1.) What would be the value of the bond described in Part d if, just after it had been issued, the expected inflation rate rose by

3percentage points, causing investors to require a

13percent return? Would we now have a discount or a premium bond? We could simply go to the input data section shown above, change the value for r from 10% to

13%. You can set up a data table to show the bond’s value at a range of rates, i.e., to show the bond’s sensitivity to changes in interest rates. This is done below. To make the data table, first type the headings, then type the rates in cells in the left column. Since the input values are listed down a column, type the formula in the row above the first value and one cell to the right of the column of values (this is B

73; note that the formula in B73 actually just refers to the bond pricing formula above in B

60). Select the range of cells that contains the formulas and values you want to substitute (A73:B78). Then click Data, What-If-Analysis, and then Data Table to get the menu. The input data are in a column, so put the cursor on “column input cell” and enter the cell with the value for r (B37), then Click OK to complete the operation and get the table. Bond Value Going rate, r: $0 0%

$0.00 7% $0.00

10% $0.00

13% $0.00

20

%

$0.00

We can use the data table to construct a graph that shows the bond’s sensitivity to changing rates.
Put B37 here.
(2.) What would happen to the value of the 10-year bond over time if the required rate of return remained at 13 percent, or if it remained at 7 percent? Would we now have a premium or a discount bond in either situation? You pick a rate.
Value of Bond in Given Year:
N 7% 10% 13%

0

1

2

3

4
5

6

7

8

9

10

You pick the rate for a bond: Your choice: 20%Resulting bond prices

$1,000

If rates fall, the bond goes to a premium, but it moves towa rds par as maturity approaches. The reverse hold if rates rise and the bond sells at a discount. If the going rate remains equal to the coupon rate, the bond will continue to sell at par. Note that the above graph assumes that interest rates stay constant after the initial change. That is most unlikely–interest rates fluctuate, and so do the prices of outstanding bonds. Yield

to

Maturity

(

YTM

)

f. (1.) What is the yield to maturity on a 10-year, 9 percent annual coupon, $1,000 par value bond that sells for $887.00? That sells for $1,134.20? What does the fact that a bond sells at a discount or at a premium tell you about the relationship between rd and the bond’s coupon rate? What is the yield-to-maturity of the bond? Use the

Rate function to solve the problem.
Years to Mat: 10

Coupon rate:
9%
Annual Pmt:

$887.00

Par value = FV:
$

s The current yield is the annual interest payment divided by the bond’s current price. The current yield provides information regarding the amount of cash income that a bond will generate in a given year. However, it does not account for any capital gains or losses that will be realized if the bond is held to maturity or call. Simply divide the annual interest payment by the price of the bond. Even if the bond made semiannual payments, we would still use the annual interest. Par value

$1,000.00 Coupon rate: 9% Current Yield =
Annual Pmt: $90.00

Current price: $887.00

YTM: 10.91% The current yield provides information on a bond’s cash return, but it gives no indication of the bond’s total return. To see this, consider a zero coupon bond. Since zeros pay no coupon, the current yield is zero because there is no interest income. However, the zero appreciates through time, and its total return clearly exceeds zero. YTM =Current Yield

+ Capital Gains Yield

Capital Gains Yield =

YTM

– Current Yield

Capital Gains Yield = –

Capital Gains Yield =

g. How does the equation for valuing a bond change if semiannual payments are made? Find the value of a 10-year, semiannual payment, 10 percent coupon bond if nominal rd = 13%.
Bonds with Semiannual Coupons
Since most bonds pay interest semiannually, we now look at the valuation of semiannual bonds. We must make three modifications to our original valuation model: (1) divide the coupon payment by 2, (2) multiply the years to maturity by 2, and (3) divide the nominal interest rate by 2.
Use the Rate function with adjusted data to solve the problem.
Periods to maturity = 10*2 =
20

Christopher Buzzard: N=20, because of semi-annual compounding (10*2 =

).

Coupon rate: 10%

Semiannual pmt = $100/2 = $50.00Bart Kreps: PMT=$50, because of semiannual payments

(100 ÷ 2) = 50
PV =
Future Value:

$1,000.00

Periodic rate = 13%/2 =
6.

Christopher Buzzard: I=6.5%, because of semi-annual compounding (13%/2 = 6.5%). Note that the bond is now more valuable, because interest payments come in faster. Excel Bond Functions Suppose today’s date is January 1, 20

14, and the bond matures on December 31, 20

23 Settlement (today) 1/1/14 Maturity 12/31/23 Coupon rate

1 0.00% Going rate, r 1 3.00% Redemption (par value) 100

Frequency (for semiannual)

2

Basis (360 or 365 day year)

0

Value of bond =

or

Notice that you could choose a current date that is between coupon payments, and the PRICE function will calculate the correct price. See the example below.
Settlement (today)

/14

Maturity 12/31/23

Coupon rate 10.00%

Going rate, r 13.00%

Redemption (par value) 100

Frequency (for semiannual) 2

Basis (360 or 365 day year) 0

Value of bond = or

This is the value of the bond, but it does not include the accrued interest you would pay. The ACCRINT function will calculate accrued interest, as shown below. Issue date 1/1/14

First interest date
6/30/14

Settlement (today) 3/25/14

Maturity 12/31/23

Coupon rate 10.00%

Going rate, r 13.00%

Redemption (par value) 100

Frequency (for semiannual) 2

Basis (360 or 365 day year) 0

or

Suppose the bond’s price is $1,

0. You can also calculate the yield using the YIELD function, as shown below. Curent price $ 1,150.00

Settlement (today) 1/1/14

Maturity 12/31/23

Coupon rate 10.00%

Redemption (par value) 100

Frequency (for semiannual) 2

Basis (360 or 365 day year) 0

h. Suppose a 10-year, 10 percent, semiannual coupon bond with a par value of $1,000 is currently selling for $1,135.90

, producing a nominal yield to maturity of 8 percent. However, the bond can be called after 5 years for a price of $1,050. (1.) What is the bond’s nominal yield to call (YTC)? (2.) If you bought this bond, do you think you would be more likely to earn the YTM or the YTC? Why? Yield to Call The yield to call is the rate of return investors will receive if their bonds are called. If the issuer has the right to call the bonds, and if interest rates fall, then it would be logical for the issuer to call the bonds and replace them with new bonds that carry a lower coupon. The yield to call (YTC) is found similarly to the YTM. The same formula is used, but years to maturity is replaced with years to call, and the maturity value is replaced with the call price.

Use the Rate function to solve the problem.

Number of semiannual periods to call: 10

Seminannual coupon rate:

5%

Semiannual Rate = I = YTC = Seminannual Pmt:$50.00

Annual nominal rate = Current price: $1,135.90Call price = FV $1,050.00

Par value $1,000.00

i. Write a general expression for the yield on any debt security (rd) and define these terms: real risk-free rate of interest (r*), inflation premium (IP), default risk premium (DRP), liquidity premium (LP), and maturity risk premium (MRP). Answer: j. Define the real risk-free rate (r*). What security can be used as an estimate of r*? What is the nominal risk-free rate (rRF)? What securities can be used as estimates of rRF? Answer: k. Describe a way to estimate the inflation premium (IP) for a T-Year bond. Answer: l. What is a bond spread and how is it related to the default risk premium? How are bond ratings related to default risk? What factors affect a company’s bond rating? Answer: m. What is interest rate (or price) risk? Which bond has more interest rate risk, an annual payment 1-year bond or a 10-year bond? Why?

Interest Rate

Risk is the risk of a decline in a bond’s price due to an increase in interest rates.

Price

sensitivity to interest rates is greater (1) the longer the maturity and (2) the smaller the coupon payment. Thus, if two bonds have the same coupon, the bond with the longer maturity will have more interest rate sensitivity, and if two bonds have the same maturity, the one with the smaller coupon payment will have more interest rate sensitivity.

Your Choice of Maturity 10-Yr Maturity 1-Yr Maturity Years to Mat: 10 Rate Price Rate Price Rate PriceCoupon rate: 10% $966.65 $946.77 $991.88 Annual Pmt: $100.00 5.0% 1,2 16

.47

5.0%

$1,386.09

5.0%

$1,047.62 Current price: $946.77 7.0% 1,123.017.0%

$1, 210.71

7.0%

$1,0 28.04 Par value = FV: $1,000.00

10.0% 1,000.00 10.0% $1,000.00 10.0% $1,000.00

YTM =

13.0%

$837.2113.0%

$973.45 15.0% 832.3915.0%

$749.06

15.0%

$956.52 Years to Mat: 1 Scratch sheet for Your Choice Coupon rate: 10% Years to Mat: 5Annual Pmt: $100.00 Coupon rate: 10%

Current price: $991.88 Annual Pmt: $100.00

Par value = FV: $1,000.00 Current price: $966.65

YTM = 10.9% Par value = FV: $1,000.00

YTM = 10.9%

Enter your choice for years to maturity:

5

As the interst rate goes from 5% to 15%, the price changes are bigger for the 10-year bond. rd 10-Year P Change 1-Year P Change

5.0%

$1,386

4.

4.5% 33.5% 15.0% $957

$749

n. What is reinvestment rate risk? Which has more reinvestment rate risk, a 1-year bond or a 10-year bond? Answer:
o. How are interest rate risk and reinvestment rate risk related to the maturity risk premium? Answer:
p. What is the term structure of interest rates? What is a yield curve? Answer:
The term structure describes the relationship between long-term and short-term interest rates. Graphically, this relationship can be shown in what is known as the yield curve. See the hypothetical curve below.
Hypothetical Inputs
See to right for actual date used in graph.
Suppose most investors expect the inflation rate to be 5 percent next year, 6 percent the following year, and 8 percent

The real risk-free rate is 3 percent. The maturity risk premium is zero for securities that mature in 1 year or less, 0.1 percent for 2-year securities, and then the MRP increases by 0.1 percent per year thereafter for 20 years, after which it is stable. What is the interest rate on 1-year, 10-year, and 20-year

Treasurysecurities? Draw a yield curve with these data. What factors can explain why this constructed yield curve is upward sloping? Real risk free rate

3.00%

Expected inflation of

5%

for the next1

years. Expected inflation of 6% for the next 1 years.Expected inflation of 8% thereafter.

Now, we want to set up a table that encompasses all of the information for our yield curve. INPUT DATA

Real risk free rate 3.00%

Expected inflation of 5% for the next 1 years.

Expected inflation of 6% for the next 1 years.

Expected inflation of 8% thereafter.

Treasury

Maturity

Yield

1 3.00%

0.00%

8.00% 2 3.00% 5.50% 0.10% 8.60% 3 3.00% 6.33% 0.20% 9.53% 4 3.00% 6.75% 0.30% 10.05% 5 3.00% 7.00% 0.40% 10.40% 6 3.00% 7. 17% 0.50% 10.67% 7 3.00%

7. 29% 0.60% 10.89% 8 3.00%

7.38% 0.70% 11.08% 9 3.00%

7.44% 0.80% 11. 24% 10 3.00%

7.50% 0.90% 1 1.40% 11 3.00% 7.55% 1.00% 11.55% 12 3.00% 7.58% 1.10% 11.68% 13 3.00% 7.62% 1.20% 11.82% The yield is upward sloping due to increasing expected inflation and an increasing maturity risk premium14 3.00%

7.64% 1.30% 11.94% 15 3.00% 7.67%1.40%

12.07% 16 3.00% 7.69% 1.50% 12. 19% q. Briefly describe bankruptcy law. If a firm were to default on the bonds, would the company be immediately liquidated? Would the bondholders be assured of receiving all of their promised payments? Answer:

17 3.00%

7.71% 1.60% 12.31% 183.00%

7.72% 1.70% 12.42% 19 3.00% 7.74% 1.80% 12.54% 20 3.00% 7.75% 1.90% 12.65% 21 3.00% 7.76% 2.00% 12.76% 223.00%

7.77% 2.10% 12.87% 23 3.00% 7.78% 2.20% 12.98% 24 3.00% 7.79% 2.30% 13.09% 25 3.00% 7.80% 2.40% 13.20% 263.00%

7.81% 2.50% 13.31% 273.00% 7.81%

2.60% 13.41% 28 3.00% 7.82% 2.70% 13.52% 29 3.00% 7.83% 2.80% 13.63% 30 3.00% 7.83% 2.90% 13.73% The table above gives us all of the components for our Treasury yield curve. Recall, we have said that Treasury securities are subject to two kinds of risk premiums, the inflation premium and the maturity risk premium. Just as we “built” Treasury yields in the table, we can “build” a yield curve based upon these expectations.

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Value of the bond over time

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000

Years to maturity

Price

10 Yr. versus 1 Yr.

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Hypothetical Treasury Yield Curve

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Value of the bond over time

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

10 Yr. versus 1 Yr.

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Hypothetical Treasury Yield Curve

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Value of the bond over time

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

10 Yr. versus 1 Yr.

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Hypothetical Treasury Yield Curve

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Value at 7%

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Value of the bond over time

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

10 Yr. versus 1 Yr.

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Hypothetical Treasury Yield Curve

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Value at 13%

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

Interest Rate Sensitivity of a 10-Year Bond

0 7.0000000000000007E-2 0.1 0.13 0.2 0 0 0 0 0

Rates fall to 7% Rates stay the same Rates increase to 13% Your choice 1000 Years to maturity

Price

0.05 7.0000000000000007E-2 0.1 0.13 0.15 1047.6190476190475 1028.0373831775701 999.99999999999989 973.45132743362842 956.52173913043487 0.05 7.0000000000000007E-2 0.1 0.13 0.15 1386.0867464592407 1210.7074462279782 1000.0000000000001 837.21269572141352 749.06156870728876 Your Choice 1216.473833531541 1123.0059230784277 1000.0000000000001 894.48306215371883 832.39224509942994 YTM

Real Risk Free Rate 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 Inflation Premium 0.05 5.5E-2 6.3333333333333339E-2 6.7500000000000004E-2 6.9999999999999993E-2 7.166666666666667E-2 7.2857142857142856E-2 7.3749999999999996E-2 7.4444444444444452E-2 7.4999999999999997E-2 7.5454545454545455E-2 7.5833333333333336E-2 7.6153846153846155E-2 7.6428571428571429E-2 7.6666666666666675E-2 7.6875000000000013E-2 7.7058823529411763E-2 7.7222222222222234E-2 7.7368421052631586E-2 7.7499999999999999E-2 Maturity Risk Premium 0 1E-3 2E-3 3.0000000000000001E-3 4.0000000000000001E-3 5.0000000000000001E-3 6.0000000000000001E-3 7.0000000000000001E-3 8.0000000000000002E-3 9.0000000000000011E-3 0.01 1.0999999999999999E-2 1.2E-2 1.3000000000000001E-2 1.4E-2 1.4999999999999999E-2 1.6E-2 1.7000000000000001E-2 1.8000000000000002E-2 1.9E-2 Maturity

Interest Rate

td>> Build a Model

SolutionSolution

6/1

5

year,

semiannual coupon bond with a par value of

may be called in 5 years at a call price of

. The bond sells for

. (Assume that the bond has just been issued.)

20

2

:

8% $1,000

$1,100

:

$1,040

5

– Hint: Write formula in words.

Cap. Gain/loss yield = (Answer)

This is a nominal rate, not the effective rate. Nominal rates are generally quoted.

8%

$0.00

8%

Coupon rate Call price

1

7/ | 1 | 5 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Chapter: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Problem: | 24 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

A | 20 | – | 8% | $1,000 | $1,040 | $1,100 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Basic Input Data: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Years to maturity: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Periods per year: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Periods to maturity: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Coupon rate | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Par value: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Periodic payment: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Current price | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Call price | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Years till callable: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Periods till callable: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

a. What is the bond’s yield to maturity? | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Peridodic YTM = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Annualized Nominal YTM = | Hint: | This is a nominal rate, not the effective rate. Nominal rates are generally quoted. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

b. What is the bond’s current yield? | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Current yield = | Hint: Write formula in words. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Hint: Cell formulas should refer to Input Section | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

(Answer) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

c. What is the bond’s capital gain or loss yield? | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Cap. Gain/loss yield = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Note that this is an economic loss, not a loss for tax purposes. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

d. What is the bond’s yield to call? | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Here we can again use the Rate function, but with data related to the call. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Peridodic YTC = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Annualized Nominal YTC = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

The YTC is lower than the YTM because if the bond is called, the buyer will lose the difference between the call price and the current price in just 4 years, and that loss will offset much of the interest imcome. Note too that the bond is likely to be called and replaced, hence that the YTC will probably be earned. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

NOW ANSWER THE FOLLOWING NEW QUESTIONS: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

e. How would the price of the bond be affected by changing the going market interest rate? (Hint: Conduct a sensitivity analysis of price to changes in the going market interest rate for the bond. Assume that the bond will be called if and only if the going rate of interest falls below the coupon rate. That is an oversimplification, but assume it anyway for purposes of this problem.) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Nominal market rate, r: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Value of bond if it’s not called: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Value of bond if it’s called: | The bond would not be called unless r | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

We can use the two valuation formulas to find values under different r’s, in a 2-output data table, and then use an IF | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

statement to determine which value is appropriate: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Value of Bond If: | Actual value, | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Not called | Called | considering | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Rate, r | $0.00 | call likehood: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

0% | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

2% | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

4% | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

6% | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

10% | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

12% | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

14% | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

16% | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

f. Now assume the date is 10/25/2014. Assume further that a 12%, 10-year bond was issued on 7/1/2014, pays interest semiannually (January 1 and July 1), and sells for $1,100. Use your spreadsheet to find the bond’s yield. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Refer to this chapter’s Tool Kit for information about how to use Excel’s bond valuation functions. The model finds the price of a bond, but the procedures for finding the yield are similar. Begin by setting up the input data as shown below: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Basic info: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Settlement (today) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Maturity | Call date | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Current price (% of par) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Redemption (% of par value) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Frequency (for semiannual) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Basis (360 or 365 day year) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Yield to Maturity: | Hint: Use the Yield function.For dates, either refer to cells D122 and D123, or enter the date in quotes, such as “10/25/2014”. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

To find the yield to call, use the YIELD function, but with the call price rather than par value as the redemption | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Yield to call: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

You could also use Excel’s “Price” function to find the value of a bond between interest payment dates. |

## Sheet2

## Sheet3

<td><

td><

h

2>Mini

Case 1 0

/28/1

5Chap

t

er

7Mini Case

SituationYour employer

,

a mid

–

sized human resources mana

g

ement company, is considering expansion into related fields, including the acquisition of Temp Force Company, an employment agency that supplies word processor

op

erators and computer programmers to businesses with temporary heavy workloads. Your employer is also considering the purchase of Biggerstaff & McDonald

(

B&M

)

, a privately held company owned by two friends, each with 5 million shares of stock. B&M currently has free cash flow of $2

4million, which is expected to grow at a constant rate of

5%. B&M’s financial statements report short-term investments of $

100 million, debt of $200 million, and preferred stock of $50 million. B&M’s weighted average cost of capital (

WACC) is

11%. Answer the following questions.

Note: There are a couple qualitative responses for this mini-case, but the remainder of the case is already computed. Why? This mini-case is similar to the normal class project and is for detailed study. Please review all aspects of this mini-case in detail in preparation for something similar in the near future. a. Describe briefly the legal rights and privileges of common stockholders. Features of Common Stock1. Common Stock represents ownership. 2. Ownership implies control.

3. Stockholders elect directors. 4. Directors hire management who attempt to maximize stock price.

Classified Stock Classified Stock carries special provisions. For example, shares could be classified as founders’ shares which come with voting rights but dividend restrictions.b. What is free cash flow (

FCF)? What is the weighted average cost of capital? What is the free cash flow valuation model? Answer:

c. Use a pie chart to illustrate the sources that comprise a hypothetical company’s total value. Using another pie chart, show the claims on a company’s value. How is equity a residual claim? Answer: Data for charts Column1 10Mkt. Sec.

1

Claims on

V

alue

Pref. Stk. 1

Debt

3

7

d. Suppose the free cash flow at Time 1 is expected to grow at a constant rate of g

L

forever. If

gL< WACC, what is a formula for the present value of expected free cash flows when discounted at the WACC? If the most recent free cash flow is expected to grow at a constant rate of gL

forever (and gL < WACC), what is a formula for the present value of expected free cash flows when discounted at the WACC?
If constant growth begins at Time 1:
If constant growth begins at Time 0:
e. Use B&M’s data and the free cash flow valuation model to answer the following questions.
INPUT DATA SECTION: Data used for valuation (in millions)
Free cash flow
$24.0
WACC 11%

Growth

5%

Short-term investments
$100.0
Debt

Vop

=

FCF1= FCF0 (1

+gL)

(WACC-gL) (WACC-gL)Vop = $25.2 0.06 Vop = $420.0

0 (2) What is its estimated total corporate value? Value of Operation

$420.0

Plus Value of Non-operating Assets

$100.0

Total Corporate Value
$520.0
(3) What is its estimated intrinsic value of equity?
Debt holders have the first claim on corporate value. Preferred stockholders have the next claim and the remaining is left to common stockholders.

Total Corporate Value $520.0

Minus Value of Debt $200.0

Minus Value of Preferred Stock

$50.0

Intrinsic Value of

Equity

$270.0 (4) What is its estimated intrinsic stock price per share? Intrinsic Value of Equity $270.0

Divided by number of shares

10.0

Intrinsic price per share
$27.00
Estimating the Value of R&R’s Stock Price (Millions, Except for Per Share Data)
INPUTS:
Value of operations
=
$420.00
Value of nonoperating assets =
$

+ Value of nonoperating assets

100.00

Total estimated value of firm
$520.00
− Debt

200.00

− Preferred stock

50.00

Estimated value of equity
$270.00
÷

Number of shares

10.00

Estimated stock price per share =

$27.00

f. You have just learned that B&M has undertaken a major expansion that will change its expected free cash flows to −$10 million in 1 year, $20 million in 2 years, and $35 million in 3 years. After 3 years, free cash flow will grow at a rate of 5%. No new debt or preferred stock were added, the investment was financed by equity from the owners. Assume the WACC is unchanged at 11% and it that there are still has 10 million shares of stock outstanding.
(1.) What is its horizon value (i.e., its value of operations at year three)? What is its current value of operations (i.e., at time zero)?
Explicit forecast:
Year

0 1 2 3

FCF

HV

3 = Vop,3 = PV of FCF4 and beyond discounted back to Year 3

Year 0 1 2 3 4 5 … t

FCF FCF3(1+gL) FCF4(1+gL) FCFt(1+gL)HV3 ←↵

←↵ ←↵

Because free cash flows are constant from Year 4 and beyond, we can apply the constant growth model at Year 3:
The general horizon value formula is:
R&R’s explicit forecast:

Year 0 1 2 3

FCF −$10.00 $20.00 $35.00 After Year 3, gL = 5%

WACC =

11%

R&R’s horizon value:
HV3 = Vop,3 =
FCF0 (1+gL)

(WACC-gL)

HV3 = Vop,3 = $36.750 6% HV3 = Vop,3 =

$612.50 After estimating the horizon value, you can estimate the current value of operations by following these steps: (1) Find the present value of the FCFs from the explicit forecast, discounted back to Time 0 at the WACC; (2) find the present value of the horizon value, discounted back to Time 0 at the WACC; and (3) sum the PV of the FCFs and the PV of the horizon value. This sum is the present value of all future FCFfrom Time 0 to infinity, discounted back to Time 0. Therefore, this sum is the current value of operations, Vop,0.

Year 0 1 2 3 4 5 … t

FCF FCF1 FCF2 FCF3

←↵ ←↵ ←↵

FCF3(1+gL) FCF4(1+gL) FCFt(1+gL)

HV3 ←↵ ←↵ ←↵

is the PV of FCF beyond the explicit forecast

←↵ ←↵ ←↵

B&M’s

Value of Operations

(Millions of Dollars)

INPUTS:

FCF −$10.00 $20.00 $35.00

↓↓

↓FCF1 FCF2 FCF3

────── ────── ──────

(1+WACC)1
(1+WACC)2
(1+WACC)3
HV = Vop,3
FCF3(1+gL)
PVs of FCFs
−$9.009
─────────
$16.232
(WACC− gL)
$25.592
PV of HV

5

$612.50 $36.75

= ──────
= ────
Vop =
$480.67

(1+WACC)3

6.00% (2.) What is its value of equity on a price per share basis? Estimating the Value of B&M’s Stock Price (Millions, Except for Per Share Data)INPUTS:

Value of operations = $480.67

Value of nonoperating assets =

$100.00All debt =

$200.00Preferred stock =

$50.00 Number of shares of common stock = 10.00

ESTIMATING PRICE PER SHARE

+ Value of nonoperating assets 100.00

Total estimated value of firm $580.67 − Debt 200.00

− Preferred stock 50.00

÷ Number of shares 10.00

Estimated stock price per share = $33.07 g. If B&M undertakes the expansion, what percent of B&M’s value of operations at Year 0 is due to cash flows from Years 4 and beyond? Hint: use the horizon value at t = 3 to help answer this question.INPUTS:

Vop,0=

$480.67

HV3 =

$612.50

First, calculate the present value of the horizon value. Then divide the Year 0 value of operations by the present value of the horizon value. This will show what percent of value is due to cash flows occurring 4 or more years in the future.
PV of HV3

=
HV3 / (1+WACC)3
PV of HV3 = $447.85

Percent of value

due to cash flows beyond Year 3

PV of HV3

=

Vop,0

Percent of value

due to cash flows beyond Year 3

=

Show

i. Your employer also is considering the acquistion of Hatfield Medical Supplies. You have gathered the following data regarding Hatfield, with all dollars reported in millions: (1) most recent sales of

; (2) most recent total net operating capital,

OpCap=

$1,120; (3) most recent operating profitability ratio,

OP=

NOPAT/

Sales=

4.5%; and (4) most recent capital requirement ratio,

CR= OpCap/Sales = 56%. You estimate that the growth rate in sales from Year 0 to Year 1 will be

10%, from Year 1 to Year 2 will be

8%, from Year 2 to Year 3 will be 5%, and from Year 3 to Year 4 will be 5%. You also estimate that the long-term growth rate beyond Year 4 will be 5%. Assume the operating profitability and capital requirement ratios will not change. Use this information to forecast Hatfield’s sales, net operating profit after taxes (NOPAT), OpCap, free cash flow, and return on invested capital (

ROIC) for Years 1 through 4. Also estimate the annual growth in free cash flow for Years 2 through 4. The weighted average cost of capital (WACC) is

9%. How does the ROIC in Year 4 compare with the WACC? No Change Actual Forecast

Year 0 1 2 3 4

Inputs WACC 9.0% Sales $2,000OpCap $1,120

Sales growth rate

10% 8% 5% 5%

NOPAT/Sales

4.5% 4.5% 4.5% 4.5% 4.5%

OpCAP/Sales
56.0%

56.0% 56.0% 56.0% 56.0%

Forecast

Sales $2,000 $2,200 $2,376 $2,495 $2,620 NOPAT $99 $107 $112 $117.879 OpCap $1,120 $1,232 $1,331 $1,397.088 $1,466.942 FCF −$13.00 $8.360 $45.738 $48.025 Growth in FCF -164% 447.1% 5.0% ROIC 8.0% 8.0% 8.0% 8.0%

Is

< WACC/(1 +

gL)?
ROIC4 = 8.0%

WACC/(1+gL)=
8.6%
Yes, ROIC4 =< WACC/(1 + gL). Therefore, we expect that the value of operations at Year 4 (HV4) should be less than the total net operating capital at Year 4 (OpCap4).
j. What is the horizon value at Year 4? What is the value of operations at Year 0? How does the value of operations compare with the current total net operating capital?
Horizon Value:
=

< 8.57%

= WACC/(1+gL)

Horizon value ≈
$1,261

< $1,467 = OpCap at horizon Current value of operations

≈

$958 < $1,120 = OpCap at horizon k. What are value drivers? What happens to the ROIC and current value of operations if expected growth increases by 1 percentage point relative to the original growth rates (including the long-term growth rate)? What can explain this? Hint: Use

ScenarioManager. Value drivers are the inputs to the free cash flow valuation model that managers are able to influence: sales growth rates, operating profitability, capital requirements, and the cost of capital. Using the Scenario Manager, the new ROIC and value of operations are: Scenario No Change

Improve Growth g0,1 10% 11%

g1,2

8% 9%

g2,3

5% 6%

g3,4

5% 6%

gL 5% 6%

OP 4.5% 4.5%

CR 56.0% 56.0%

ROIC 8.0% 8.0%

Current value of operations $958

9.00%

WACC/(1+WACC)
8.26%

8.26%

Growth hurts value because the ROIC is too low. Growth will only help value if ROIC>WACC/(1+WACC).
l. Assume growth rates are at their original levels. What happens to the ROIC and current value of operations if the operating profitability ratio increases to

? Now assume growth rates and operating profitability ratios are at their original levels. What happens to the ROIC and current value of operations if the capital requirement ratio decreases to 51%? Assume growth rates are at their original levels. What is the impact of simultaneous improvements in operating profitability and capital requirements? What is the impact of simultaneous improvements in the growth rates, operating profitability, and capital requirements? Hint: Use Scenario Manager. Using the Scenario Manager and improving operating profitability, the new ROIC and value of operations are: Scenario No Change

Improve OP g0,1 10% 10%g1,2 8% 8%

g2,3 5% 5%

g3,4 5% 5%

gL 5% 5%

OP 4.5% 5.5%

CR 56.0% 56.0%

ROIC 8.0% 9.8% Current value of operations $958 $1,523 WACC 9.00% 9.00%

WACC/(1+WACC) 8.26% 8.26%

g0,1 10% 10%

g1,2 8% 8%

g2,3 5% 5%

g3,4 5% 5%

gL 5% 5%

OP 4.5% 4.5%

WACC 9.00% 9.00%

WACC/(1+WACC) 8.26% 8.26%

g0,1 10% 10%

g1,2 8% 8%

g2,3 5% 5%

g3,4 5% 5%

gL 5% 5%

OP 4.5% 5.5%

CR 56.0% 51.0%

WACC 9.00% 9.00%

WACC/(1+WACC) 8.26% 8.26%

g0,1 10% 11%

g1,2 8% 9%

g2,3 5% 6%

g3,4 5% 6%

gL 5% 6%

OP 4.5% 5.5%

CR 56.0% 51.0%

ROIC 8.0% 10.8%

WACC 9.00% 9.00%

WACC/(1+WACC) 8.26% 8.26%

gL 8.0% 8.8% 9.8% 10.8%

5% $958 $1,191 $1,523 $1,756

6% $933 $1,247 $1,694

$2,008

Notice that small changes in ROIC and growth cause large changes in value.
n. (1.) Write out a formula that can be used to value any dividend-paying stock, regardless of its dividend pattern.
The value of any financial asset is equal to the present value of future cash flows provided by the asset. When an investor buys a share of stock, he or she typically expects to receive cash in the form of dividends and then, eventually, to sell the stock and to receive cash from the sale. Moreover, the price any investor receives is dependent upon the dividends the next investor expects to earn, and so on for different generations of investors. Thus, the stock’s value ultimately depends on the cash dividends the company is expected to provide and the discount rate used to find the present value of those dividends.
Here is the basic dividend valuation equation:
D1

+

D2+

. . . . DN ( 1 + rs ) ( 1 + rs ) 2 ( 1 + rs ) N The dividend stream theoretically extends on out forever, i.e., n = infinity. Obviously, it would not be feasible to deal with an infinite stream of dividends, but fortunately, an equation has been developed that can be used to find the PV of the dividend stream, provided it is growing at a constant rate. Naturally, trying to estimate an infinite series of dividends and interest rates forever would be a tremendously difficult task. Now, we are charged with the purpose of finding a valuation model that is easier to predict and construct. That simplification comes in the form of valuing stocks on the premise that they have a constant growth rate. n. (2.) What is a constant growth stock? How are constant growth stocks valued? In this stock valuation model, we first assume that the dividend and stock will grow forever at a constant growth rate. Naturally, assuming a constant growth rate for the rest of eternity is a rather bold statement. However, considering the implications of imperfect information, information asymmetry, and general uncertainty, perhaps our assumption of constant growth is reasonable. It is reasonable to guess that a given firm will experience ups and downs throughout its life. By assuming constant growth, we are trying to find the average of the good times and the bad times, and we assume that we will see both scenarios over the firm’s life. In addition to assuming a constant growth rate, we will be estimating a long-term required return for the stock. By assuming these variables are constant, our price equation for common stock simplifies to the following expression: D1( rs – gL ) In this equation, the long-run growth rate (g) can be approximated by multiplying the firm’s return on assets by the retention ratio. Generally speaking, the long-run growth rate of a firm is likely to fall between 5% and 8% a year. n. (3.) What happens if a company has a constant gL which exceeds rs? Will many stocks have expected growth greater than the required rate of return in the short run (i.e., for the next few years)? In the long run (i.e., forever)? Answer: See Chapter 7 Mini Case Show. o. Assume that Temp Force has a beta coefficient of 1.2, that the risk-free rate (the yield on T-bonds) is 7.0%, and that the market risk premium is 5%. What is the required rate of return on the firm’s stock? CAPM = rRF + b (rRF – rM) 7% + 1.2(5%) = 13% p. Assume that Temp Force is a constant growth company whose last dividend (D0, which was paid yesterday) was $2.00

and whose dividend is expected to grow indefinitely at a 6% rate. (1.) What is the firm’s current stock price? (2.) What is the stock’s expected value 1 year from now? (3.) What are the expected dividend yield, the capital gains yield, and the total return during the first year? Constant Growth Model:

INPUTS:

D0 = $2.00

gL = 6%

rs =
13.0%
D1 =

D1 = D0 (1 + gL) = $2.12 P0

=

D1 = $2.12

( rs – gL )

D2

( rs – gL )

D2 = D1 (1+gL) = $2.2472 P1 = $2.2472

0.07

P1 = $32.10 DividendYield

=

D1

CG Yield = P1 – P0 P0 P0Dividend Yield =

$2.12 CG Yield =

$1.82 $30.29 $30.29Dividend Yield = 7.00%

CG Yield =

6.00%Bart Kreps: For a constant growth stock, the capital gains yield equals the growth rate. Total Yield =

Dividend Yield

+ CGYield Total Yield = 13.00% q. Now assume that the stock is currently selling at $30.29. What is its expected rate of return? Rearrange to rate of return formula D1 + gL

P0

$2.12 + 0.06$30.29

13%r. Now assume that Temp Force’s dividend is expected to experience nonconstant growth of 30%

from Year 0 to Year 1,

25%from Year 1 to Year 2, and

15%from Year 2 to Year 3. After Year 3, dividends will grow at a constant rate of 6%. What is the stock’s intrinsic value under these conditions? What are the expected dividend yield and capital gains yield during the first year? What are the expected dividend yield and capital gains yield during the fourth year (from Year 3 to Year 4)? For many companies, it is unreasonable to assume that it grows at a constant growth rate. Hence, valuation for these companies proves a little more complicated. The valuation process, in this case, requires us to estimate the short-run non-constant growth rate and predict future dividends. Then, we must estimate a constant long-term growth rate at which the firm is expected to grow. Generally, we assume that after a certain point of time, all firms begin to grow at a rather constant rate. Of course, the difficulty in this framework is estimating the short-term growth rate, how long the short-term growth will hold, and the long-term growth rate. Specifically, we will predict as many future dividends as we can and discount them back to the present. Then we will treat all dividends to be received after the convention of constant growth rate with the Gordon constant growth model described above. The point in time when the dividend begins to grow at a constant rate is called the horizon date. When we calculate the constant growth dividends, we solve for a horizon value (also called the terminal value or continuing value) as of the horizon date. We can then find the present value of the dividends in the forecast period and the present value of the horizon value, which gives the current estimated stock price. Process for Finding the Value of a Nonconstant Growth Stock

INPUTS:

D0 = $2.00 Last dividend the company paid. rs = 13.0% Stockholders’ required return. g0,1 =30%

Growth ratefor Year 1 only. g1,2 =

25%

Growth rate for Year 2 only. g2,3 =15%

Growth rate for Year 3 only. gL = 6% Constant long-run growth rate for all years after Year 3. Growth rate 30% 25% 15% 6% 6%Year 0 1 2 3 4

Dividends $2.6000 $3.2500 $3.7375↓ ↓ ↓

D1 D2 D3 D4 ────── ────── ────── ──── = (1+rs)1 (1+rs)2 (1+rs)3 (rs− gL)↓

D3 (1+gL) PVs of dividends $2.301 ────── = $2.545 (rs− gL)

$2.590

↓

PV of HV3
$39.224
$56.596
$3.962
= ───────

$56.596

= ──── =

$46.661

(1+rs)3 7.00%

Expected Dividend and CG Yields at t = 0
Dividend Yield =

13.0%

Expected Dividend and CG Yields at t = 3
Dividend Yield =

Total Return = 13.0%

s. What is the market multiple method of valuation? What are its strengths and weaknesses? Answer: See Chapter 7 Mini Case Show t. What are the advantages of the free cash flow valuation model relative to the dividend growth model? Answer: See Chapter 7 Mini Case Show u. What is preferred stock? Suppose a share of preferred stock pays a dividend of $2.10and investors require a return of 7%. What is the estimated value of the preferred stock? The dividend stream would be a perpetuity. Vps =

Dividend ÷

rps Vps = $2.10 ÷ 7.00%Vps = $30.00

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

Value of Operations

Column1

Mkt. Sec. 10 1

Equity

Claims on Value

Pref. Stk. Debt 1 3 7

## Scenario Summary

Scenario SummaryNo Change Improve Growth Improve OP Improve CR Improve All Improve OP and CR

Modified by Mike Ehrhardt on 5/30/2014

Created by Mike Ehrhardt on 5/30/2014

Modified by Mike Ehrhardt on 5/30/2014 Created by Mike Ehrhardt on 5/30/2014 Created by Mike Ehrhardt on 5/30/2014 Created by Mike Ehrhardt on 5/30/2014 Created by Mike Ehrhardt on 5/30/2014

Modified by Mike Ehrhardt on 3/13/2015

Created by Mike Ehrhardt on 3/13/2015

No Change No Change Improve Growth Improve OP Improve CR Improve All Improve OP and CR Improve OP and Growth Improve Growth

10% 10% 11% 10% 10% 11% 10% 11% 11%

8% 8% 9% 8% 8% 9% 8% 9% 9%

5% 5% 6% 5% 5% 6% 5% 6% 6%

5% 5% 6% 5% 5% 6% 5% 6% 6%

4.5% 4.5% 4.5% 5.5% 4.5% 5.5% 5.5% 5.5% 4.5%

4.5% 4.5% 4.5% 5.5% 4.5% 5.5% 5.5% 5.5% 4.5%

4.5% 4.5% 4.5% 5.5% 4.5% 5.5% 5.5% 5.5% 4.5%

4.5% 4.5% 4.5% 5.5% 4.5% 5.5% 5.5% 5.5% 4.5%

56.0% 56.0% 56.0% 56.0% 51.0% 51.0% 51.0% 56.0% 51.0%

56.0% 56.0% 56.0% 56.0% 51.0% 51.0% 51.0% 56.0% 51.0%

56.0% 56.0% 56.0% 56.0% 51.0% 51.0% 51.0% 56.0% 51.0%

56.0% 56.0% 56.0% 56.0% 51.0% 51.0% 51.0% 56.0% 51.0%

$958 $958 $933 $1,523 $1,191 $2,008 $1,756 $1,694 $1,247

Current Values: | Improve OP and Growth | Improve CR and Growth | |||||

Created by Mike Ehrhardt on 5/30/2014 | Created by Mike Ehrhardt on 3/13/2015 | ||||||

Changing Cells: | |||||||

$A$304 | |||||||

$C$310 | |||||||

$D$310 | |||||||

$E$310 | |||||||

$F$310 | |||||||

$C$311 | |||||||

$D$311 | |||||||

$E$311 | |||||||

$F$311 | |||||||

$C$312 | |||||||

$D$312 | |||||||

$E$312 | |||||||

$F$312 | |||||||

Result Cells: | |||||||

$C$333 | |||||||

Notes: Current Values column represents values of changing cells at | |||||||

time Scenario Summary Report was created. Changing cells for each | |||||||

scenario are highlighted in gray. |

L

L

t

t

,

op

t

g

WACC

)

g

1

(

FCF

V

HV

–

+

=

=

L

Lt

t,opt

gWACC

)g1(FCF

VHV

L

1

0

,

op

g

WACC

FCF

V

–

=

L

L

0

0

,

op

g

WACC

)

g

1

(

FCF

V

–

+

=

L

L

3

3

,

op

3

g

WACC

)

g

1

(

FCF

V

HV

–

+

=

=

L

L3

3,op3

gWACC

)g1(FCF

VHV

>Build a Model

/

6/15

7

1 2

$20.0

40

Current Projected

0 1 2 3 4

Free cash flow

$20.0 $80.0 $84.0

(PV of FCF + HV)

Value of operationsSolution | 7 | 1 | ||||||||||||||||||||||||||

Chapter: | Valuation of Stocks and Corporations | |||||||||||||||||||||||||||

Problem: | 22 | |||||||||||||||||||||||||||

Selected data for the Derby Corporation are shown below. Use the data to answer the following questions. | ||||||||||||||||||||||||||||

INPUTS (In millions) | Year | |||||||||||||||||||||||||||

Current | Projected | |||||||||||||||||||||||||||

0 | 3 | 4 | ||||||||||||||||||||||||||

Free cash flow | – | $20.0 | $80.0 | $84.0 | ||||||||||||||||||||||||

Marketable Securities | $ | 40 | ||||||||||||||||||||||||||

Notes payable | $100 | |||||||||||||||||||||||||||

Long-term bonds | $300 | |||||||||||||||||||||||||||

Preferred stock | $50 | |||||||||||||||||||||||||||

WACC | 9.00% | |||||||||||||||||||||||||||

Number of shares of stock | ||||||||||||||||||||||||||||

a. Calculate the estimated horizon value (i.e., the value of operations at the end of the forecast period immediately after the Year-4 free cash flow). Assume growth becomes constant after Year 3. | ||||||||||||||||||||||||||||

-$20.0 | ||||||||||||||||||||||||||||

Long-term constant growth in FCF | ||||||||||||||||||||||||||||

Horizon value | ||||||||||||||||||||||||||||

b. Calculate the present value of the horizon value, the present value of the free cash flows, and the estimated Year-0 value of operations. | ||||||||||||||||||||||||||||

PV of horizon value | ||||||||||||||||||||||||||||

PV of FCF | ||||||||||||||||||||||||||||

Value of operations | ||||||||||||||||||||||||||||

c. Calculate the estimated Year-0 price per share of common equity. | ||||||||||||||||||||||||||||

Plus value of narketable securities | ||||||||||||||||||||||||||||

Total value of company | ||||||||||||||||||||||||||||

Less value of debt | ||||||||||||||||||||||||||||

Less value of preferred stock | ||||||||||||||||||||||||||||

Estimated value of common equity | ||||||||||||||||||||||||||||

Divided by number of shares | ||||||||||||||||||||||||||||

Price per share |