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>Mini Case 1 0

5 Chapter

. 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

) a $

lump sum cash flow at the end of

Year

2, (2) an ordinary annuity of $

100

per year for

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

5, and $50 at the end of

Years

0 through 3.

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

% Time period

0 1

2
FV

at year end Or

dinary annuity of $100 per year for three years.

I%
Time period 0 1 2
FV at year end

Uneven cash flow stream.

Time period 0 1 2 3

FV 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

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)

5% 10% 15

% 0

4
6 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 Back

Time period 0 1 2 3

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.

. 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

% 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 $

1.0

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 3
PV

-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

Mini 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

ANNUITY 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 0.1
PMT

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

CFt

Annuity PV PV3

=
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

CFt

Annuity FV FV3 =
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 =

Using the function wizard, we follow the same procedure as above, except remember to enter a “1” to tell Excel that in this problem the payments occur at the beginning of the periods.

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

300

-50 Cash Flow

s PV of Cash Flows 0

0.91 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

CFN PV0 0 0
1 100
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/yr

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>

for monthly, and

0 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

+ (10%/2))^2 –

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

0.12

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

EAR

MORTGAGE. GRAPH BELOW. THE LONGER THE MATURITY, THE SMALLER THE INITIAL PRINCIPAL PAYMENT. N 3

PMT = Total pmts Tot. int. paid Tot. prin. pd

N 30 PMT =

$106.08 I 0.1 I 0.1
PV

1000

PV 1000
Now, construct an amortization table for the loan described above.

Beg. Amt. Payment

Interest

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? (

273

days) 29 30
$0.00

$0.00 $0.00
0 1 2 3 4 5 273

100

0.00031054
Bart 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

0.5

1.5

2.5

3
Periods 0 1.0 2

3.0

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 6
PV 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

456

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

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

0.00018538 N 456
Bank account:

FV < $1,000,

so buy the note. See which has the greater present value

0 1 2 3 4 5 456

1000

I 0.00018538
N 456

PV of the note:

> $859 cost, so buy the note. See which has the higher effective rate of return, EFF%

0 1 2 3 4 5 456

850 1000

N 456

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

Value of Money

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.

. 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

. 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:

10%

N = 5
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:

I/YR = ?

N = 5

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

Inputs: PV =

FV =

N = ?

. Then find the FV of that same annuity.

Inputs:

N = 5

15%

PV =

FV =

x =

Inputs:

1000 1000

I/YR = 10% 5%
N = 5

Formula:

Wizard (FV):

Orig. Inputs New Inputs

Inputs: FV = 1000 1000

I/YR = 10% 5%
N = 5 10
Formula: PV = FV/(1+I)^N =
Wizard (PV):

Year

100
2 200
3

PV =
Year Payment x

= FV

1 100

2 200

6.00

3 400

PV =

,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.

Year

Pmt Interest

1
2
3
4
5

6
7
8
9

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?

Beg. Amt. Pmt Interest Principal End. Bal.

1
2
3
4
5
6
7
8
9
10

11
12
15
16
17
18
20
21
30
32
33
38
40
45
47
50

64
66
67
85

100

120

Solution			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:
Month
13
14
19
22
23
24
25
26
27
28
29
31
34
35
36
37
39
41
42
43
44
46
48
49
51
52
53
54
55
56
57
58
59
61
62
63
65
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
86
87
88
89
90
91
92
93
94
95
96
97
98
99
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119

>Mini Case 1 0

5 Chapter 5. Mini Case Situation Sam Strother and Shawna Tibbs are vice

–

presidents of Mutual of Seattle Insurance Company and co-direct

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

orthwestern 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

-year,

$1,000

par 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

percentage points, causing investors to require a

percent 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

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

0). 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

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

s 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

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:

$90.00 Going rate, r =

YTM: See RATE function at right. Current price:

$887.00
Par value = FV: $

1,000.00 (2.) What are the total return, the current yield, and the capital gains yield for the discount bond? (Assume the bond is held to maturity and the company does not default on the bond.) Current and

Capital Gains Yield

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 = $50.00

Bart 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

, and the bond matures on December 31, 20

23 Settlement (today) 1/1/14 Maturity

/31/23 Coupon rate

0.00% Going rate, r

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

Accrued interest =

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

Yield
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:

Semiannual Rate = I = YTC = Seminannual Pmt:

$50.00

Annual nominal rate = Current price: $1,135.90
Call 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 Price
Coupon rate: 10%

$966.65 $946.77 $991.88 Annual Pmt:

$100.00 5.0% 1,2

.47

5.0%

$1,386

.09

5.0%

$1,047.62 Current price: $946.77

7.0% 1,123.01

7.0%

$1,

0.71

7.0%

$1,0

.04 Par value = FV: $1,000.00

10.0%

1,000.00 10.0% $1,000.00 10.0% $1,000.00
YTM =

10.9% 13.0% 894.48

13.0%

$837.21

13.0%

$973.45 15.0% 832.39

15.0%

$749

.06

15.0%

$956.52 Years to Mat: 1

Scratch sheet for Your Choice Coupon rate: 10% Years to Mat: 5
Annual 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:

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%

$1,048

$1,386
4.

8% 38.

6% 10.0% $1,000 $1,000
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

thereafter.

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

Treasury

securities? 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

for the next

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.

Years to

Real risk-free Inflation Maturity Risk

Treasury
Maturity

rate (r*) Premium (IP) Premium (MRP)

Yield
1 3.00%

5.00%

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%

% 0.50% 10.67% 7 3.00%

% 0.60% 10.89% 8 3.00%

7.38% 0.70% 11

.08% 9 3.00%

7.44% 0.80% 11.

% 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 premium

14 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.

% 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% 18

3.00%

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% 22

3.00%

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% 26

3.00%

7.81% 2.50% 13.31% 27

3.00% 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