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Chapter 6

Bonds and Bond

Valuation

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1. Understand basic bond terminology and apply thetime value of money equation in pricing bonds.2. Understand the difference between annual and

semiannual bonds and note the key features ofzero-coupon bonds.

3. Explain the relationship between the couponrate and the yield to maturity (YTM).

4. Delineate bond ratings and why ratings affectbond prices.

5. Appreciate bond history and understand therights and obligations of buyers and sellers ofbonds.

6. Price government bonds, notes, and bills.

Learning Objectives

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6.1 Application of the Time Value ofMoney Tool: Bond Pricing

Bonds - Long-term debt instruments (maturity -over 1 year)

Provide periodic interest income annuity series

Return of the principal amount at maturity future

lump sum Prices can be calculated by using present value

(PV) techniques i.e. discounting of future cashflows.

Combination of PV ofan annuity and ofa lumpsum

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Table 6.1 Bond Information onJuly 15, 2008

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6.1 (A) Key Components of a Bond (continued):

Figure 6.1 Merrill Lynch corporate bond

Par value : Typically $1000

Coupon rate: Annual rate of

interest paid.

Coupon: Regular interest

payment received by holder

per year.

Maturity date: Expiration

date of bond when par value

is paid back.

Yield to maturity: Expected

rate of return based on

current market price of bond

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6.1 (B) Pricing a Bond in Steps [Figure 6.2 ]

Since bonds involve a combination ofan annuity

(coupons)and a lump sum (par value) its priceis best calculated by using the following steps:

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6.1 (B) Pricing a Bond in Steps (continued):

Example 2

Calculate the price of an AA-rated, 20-year, 8% coupon

(paid annually) corporate bond (Par value = $1,000) whichis expected to earn a yield to maturity of 10%.

Annual coupon = Coupon rate * Par value= .08 * $1,000 = $80 [+/-] [PMT]

YTM = r = 10% [I/Y]

Maturity = n = 20 [N]

Principal at maturity (par value) = FV = 1000 [+/-] [FV]

Price of bond = PV of coupons + PV of par value= 681.08 + 148.64 = 829.72

CPT [PV] 829.72

Year 0 1

$80

2

$80

3

$80

20

$80$1,000

18 19

$80 $80

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Most corporate and government bonds pay coupons on a

semiannual basis.

Some companies issue zero-coupon bonds by selling them at

a deep discount.

For computing price of these bonds, the values of the inputs

have to be adjusted according to the frequency of the coupons(or absence thereof).

For example, for semi-annual bonds, the annual coupon is

divided by 2, the number of years is multiplied by 2, and

the YTM is divided by 2.

The price of the bond can then be calculated by using the

TVM equation, a financial calculator, or a spreadsheet.

6.2 Semiannual Bonds andZero-Coupon Bonds

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6.2 Semiannual Bond ExampleFigure 6.4 Coca-Cola Semiannual Bond.

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6.2 Coca-Cola Semiannual Bonds at original issue(continued)

Using TVM Equation:

8.8% YTM at original issue

Using Financial Calculator:

30 x 2 = 60 [N] 8.8 2 = 4.4 [I/Y] at original issue 85 2 = 42.50 [+/-] [PMT] 1000 [+/-] [FV]

CPT [PV] 968.48

Figure 6.5Future cash

flow of theCoca-Colabond

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6.2 Semiannual Bonds and Zero-Coupon Bonds (continued)

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6.2 (A) Pricing Bonds after Original Issue

The price of a bond is. a function of the remaining cash flows

(i.e. coupons and par value) that would be paid on it untilexpiration

As of August, 2008, the 8.5%, 2022 Coca-Cola bond hasonly 27 coupons left to be paid on it until it matures on Feb. 1,

2022

Figure 6.6 Remaining cash flow of the Coca-Cola bond

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EXAMPLE 3 - 6.2 (A) Pricing Bonds After Original Issue

Four years ago, the XYZ Corporation issued an 8% coupon(paid semi-annually), 20-year, AA-rated bond at its par value of

$1000. Currently, the yield to maturity on these bonds is 10%.Calculate the price of the bond today.

Remaining number of semi-annual coupons

= (20 - 4) * 2 = 32 coupons [N]

Semi-annual coupon = (.08*1000)/2 = $40 [+/-] [PMT]

Par value = $1000 = [+/-] [FV]

Annual YTM = 10% YTM/2 5% = [I/Y]

CPT [PV] 841.97

Input: N I/Y PV PMT FV

Key: 32 5 ? -40 -1000Output 841.97

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6.2 (A) Pricing Bonds afterOriginal Issue (continued)

Method 1: Using TVM equations

Bond Price =

r

r1

11

Couponr1

1ValuePar

n

n

Bond Price =

0.05

0.051

11

$400.051

1$1,000

32

32

Bond Price = $1000 x 0.209866 + $40 x 15.80268Bond Price = $209.866 + $632.107Bond Price = $841.97

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6.2 (A) Pricing Bonds afterOriginal Issue (continued)

Method 2: Using a financial calculator

Input: N I/Y PV PMT FV

Key: 32 5 ? -40 -1000

Output 841.97

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6.2 (B) Zero-Coupon Bonds

Known as pure discount bonds and sold ata discount from face value

Do not pay any interest over the life of thebond.

At maturity, the investor receives the parvalue, usually $1000.

Price of a zero-coupon bond is calculated by

merely discounting its par value at theprevailing discount rate or yield to maturity.

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6.2 (C) Amortization of a Three-Year Zero-Coupon Bond w/ 8% Yield [Table 6.2]

The discount on a zero-coupon bond is amortized over its life.Interest earned is calculated for each 6-month period.

for example .04*790.31=$31.62Interest is added to price to compute the ending price.Zero-coupon bond investors have to pay tax on annual price

appreciation even though no cash is received.

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Example 4: Price of and taxesdue on a zero-coupon bond

John wants to buy a 20-year, AAA-rated, $1000par value, zero-coupon bond being sold byDiversified Industries Inc. The yield to maturity onsimilar bonds is estimated to be 9%.

4A - How much would he have to pay for it (=what is the reasonable price of this bond)?

4B - How much will he be taxed on the investment

after 1 year, if his marginal tax rate is30%?

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Example 4A Answer (continued)

Method 1: Using TVM equation

Bond Price = Par Value * [1/(1+r)n]

Bond Price = $1000*(1/(1.045)40

Bond Price = $1000 * .1719287 = $171.93

Method 2: Using a financial calculator

Input: N I/Y PV PMT FVKey: 40 4.5 ? 0 -1000

Output 171.93

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Example 4B Answer (continued)Calculate the price of the bond at the end offirst year.

19 x 2 = 38 remaining couponsInput: N I/Y PV PMT FVKey: 38 4.5 ? 0 -1000Output 187.75

Taxable income = Price at the end of first year price at the issue = $187.75 - $171.93

= $15.82

Taxes due = Tax rate * Taxable income= 0.30*$15.82

= $4.75

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Example 4B (Answer) (continued)

Alternately, we can calculate the semi-annualinterest earned, for each of the two semi-annual

periods during the year.

$171.93 * .045 = $7.736

$171.93+7.736 = $179.667 Price after 6 months

$179.667 * .045 = $8.084

$179.667+8.084 = $187.75 Price at end of year

Total interest income for 1 year

= $7.736 + $8.084 = $15.82

Tax due = 0.30 * $15.82 = $4.75

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6.3 Yields and Coupon Rates

A bonds coupon rate differs from its yieldto maturity (YTM).

Coupon rate -- set by the company at thetime of issue and is fixed (except for newerinnovations which have variable couponrates)

YTM is dependent on market, economic,

and company-specific factors and istherefore variable.

6 3 (A) Th Fi t I t t R t

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6.3 (A) The First Interest Rate:Yield to Maturity

Expected rate of return on a bond if held tomaturity.

The price that willing buyers and sellers settleat determines a bonds YTM at any given point.

Changes in economic conditions and risk factorswill cause bond prices and their correspondingYTMs to change.

YTM can be calculated by entering the couponamount (PMT), price (PV), remaining number ofcoupons (n), and par value (FV) into the TVMequation, financial calculator, or spreadsheet.

6 3 (B) Th Oth I t t

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6.3 (B) The Other InterestRate: Coupon Rate

The coupon rate on a bond is set by theissuing company at the time of issue

It represents the annual rate of interest thatthe firm is committed to pay over the life of

the bond. If the rate is set at 7%, the firm is

committing to pay .07*$1000 = $70 per

year on each bond, It is paid either in a single check or two

checks of $35 paid six months apart.

6 3 (C) R l ti hi f YTM d

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6.3 (C) Relationship of YTM andCoupon Rate An issuing firm gets the bond rated by a rating

agency such as Standard & Poors or Moodys. Then, based on the rating and planned maturity

of the bond, it sets the coupon rate to equal theexpected yield as indicated in the Yield Book

(available in the capital markets at that time)and sells the bond at par value ($1000).

Once issued, if investors expect a higher yieldon the bond, its price will go down and thebond will sell below par or as a discount bondand vice-versa.

Thus, a bonds YTM can be equal to (parbond), higher than (discount bond) or lowerthan (premium bond) its coupon rate.

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6 3 (C) R l ti hi f Yi ld t

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6.3 (C) Relationship of Yield toMaturity and Coupon Rate(continued)

Figure 6.8 Bond prices and interest rates move in oppositedirections.

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Example 5: Computing YTM

Last year, The ABC Corporation had issued 8%coupon (semi-annual), 20-year, AA-rated

bonds (Par value = $1000) to finance itsbusiness growth.

5A - If investors are currently offering $1200 on

each of these bonds, what is their expectedyield to maturity on the investment?

5B - If you are willing to pay no more than $980for this bond, what is your expected YTM?

Remaining number of coupons = 19 * 2 = 38 Semi-annual coupon amount

=( 0.08 * $1000 ) 2 = $40

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Example 5A - Answer

PV = $1200 (current market price)

TI-BAII+: 38 [N] 1200 [PV]

40 [+/-] [PMT] 1000 [+/-] [FV]

CPT [I/Y] 3.097

Thus, annual YTM = 3.097 x 2 = 6.19

Note: This is apremium bond, so its YTM(6.19%) < Coupon rate (8%)

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Example 5B - Answer (continued)

PV = $980

TI-BAII+: 38 [N]980 [PV]

40 [+/-] [PMT]

1000 [+/-] [FV]

CPT [I/Y] 4.1048

Thus, the annual YTM = 4.1048 x 2 = 8.2

Note: This would be a discount bond, so its YTM(8.2%) > Coupon rate (8%)

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6.4 Bond Ratings

Ratings are produced by Moodys, Standard andPoors, and Fitch

Range from AAA(top-rated) to C (lowest-rated) or D(default).

Help investors gauge likelihood of default by issuer.

Assist issuing companies establish a yield on newly-issued bonds. Junk bonds: is the label given to bonds that are rated below BBB.

These bonds are considered to be speculative in nature and carryhigher yields than those rated BBB or above (investment grade).

Fallen angels: is the label given to bonds that have had theirratings lowered from investment to speculative grade.

Credibility Issue: 2008 Financial Crisis revealed seriouscorruption and incompetency of the rating agencies, whichled the U.S. Department of Justice to sue them recently

[2013.02]

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Table 6.4Bond Ratings

6 5 Some Bond History and More

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6.5 Some Bond History and MoreBond Features

Corporate bond features have gone throughsome major changes over the years.

Bearer bonds:

Indenture or deed of trust:

Collateral:

Mortgaged security:

Debentures:

Senior debt: Sinking fund:

Protective covenants:

6 5 Some Bond History and More

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6.5 Some Bond History and MoreBond Features (continued)

Callable bond: Yield to call:

Putable bond:

Convertible bond:

Floating-rate bond:

Prime rate:

Income bonds:

Exotic bonds:

Example 6: Calculating Yield to

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Example 6: Calculating Yield toCall

Two years ago, the Mid-AtlanticCorporation issued a 10% coupon (paidsemi-annually), 20-year maturity, bond witha 5-year deferred call feature and a call

penalty of one coupon payment in additionto the par value ($1000) if exercised.

If the current price on these bonds is

$1080, what is its yield to call?

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Example 6 Answer Yield to Call

Remaining number of coupons until first call date

= 5 (5-year deferred call feature)2 (2 years passed)= 3 years = 3 x 2 = 6 [N]

Semi-annual coupon = 1000 x 10% = $50[+/-] [PMT]

Call price = Par value + penalty (one coupon in this case)

= 1000 + 50= $1050[+/-] [FV]

Current market price = $1080[PV]

CPT [I/Y] 4.2131Therefore, the annual Yield to Call (YTC)= 4.2131 x 2 = 8.43%

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6.6 U.S. Government Bonds Include bills, notes, and bonds sold by the

Department of the Treasury

State bonds, issued by state governments

Municipal bonds issued by county, city, or localgovernment agencies.

Treasury bills, are zero-coupon, pure discountsecurities with maturities ranging from 1-, 3-, and6-months up to 1 year.

Treasury notes have between two to 10 yearmaturities.

Treasury bonds have greater than 10-yearmaturities, when first issued.

6 6 U S (Federal) Government

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6.6 U.S. (Federal) GovernmentBonds (continued)

Table 6.6 Government Notes and Bonds,Prices as of April 8, 2008

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6.6 (A) Pricing a U.S. Government Bond Similar to the method used for pricing corporate bonds and can be

done by using TVM equations, a financial calculator or a spreadsheetprogram.

For example, lets assume you are pricing a 7-year, 6% coupon(semi-annual) $100,000 face value Treasury note, using anexpected yield of 8%:

Figure 6.11 U.S.Government Treasurynote cash flows.

TI-BAII+: 7 x 2 = 14 [N]

8 2 = 4 [I/Y]

100,000 x 0.06 2 = 3,000[+/-] [PMT]

100,000[+/-] [FV]

CPT

[PV] 89,436.88

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6.6 (B) Pricing a Treasury bill

Price of T-bill

= Face value * [1 - BDY * DTM 360]DTM= Days to maturity

Bank discount yield (BDY)is a special discount rate used

in conjunction with treasury bills under a 360 day-per-yearconvention (commonly assumed by bankers).

Bond equivalent yield (BEY)is the APR equivalent of thebank discount yield calculated by adjusting it as follows:

BEY= 365 * BDY________360 - (DTM * BDY)

6 6 (B) Pricing a Treasury bill

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6.6 (B) Pricing a Treasury bill(continued)

Table 6.7 Selected Historical Treasury Bill BankDiscount Rates

Example 7: Calculating the price and BEY

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Example 7: Calculating the price and BEYof a Treasury billCalculate the price and BEY of a treasury bill which matures in

105 days, has a face value of $10,000 and is currently beingquoted at a bank discount yield of 2.62%.

Price of T-bill= Face value * [1 - BDY * DTM 360]= 10000 x [ 1 (0.0262) x 105 360 ]

= 10000 x 0.9923583= $ 9,923.58

BEY= 365 * BDY________

360 - (DTM * BDY)

= [ 365 x 0.0262 ] [ 360 ( 105 x 0.0262 ) ]= 9.563 357.249= 0.0268= 2.68%

Additional Problems 1:

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Additional Problems 1:Pricing a semi-annual bond

Last year, The Harvest Time Corporation sold$40,000,000 worth of7.5% coupon, 15-year maturity, $1000 par value, AA-rated;non-callable bonds to finance its business

expansion. Currently, investors aredemanding a yield of 8.5% on similar bonds.If you own one of these bonds and want tosell it, how much money can you expect to

receive on it (= what is the reasonable price)?

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Additional Problems 1 (Answer)

Using a financial calculator

TI-BAII+: 15 1 = 14 years remaining

14 x 2 = 28 [N]

8.5 2 = 4.25 [I/Y]

1000 x 0.075 2 = 37.5[+/-] [PMT]

1000[+/-] [FV]

CPT [PV] 919.03


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Additional Problems 2:Yield-to-Maturity

Joe Carter is looking to invest in a four-yearbond that pays semi-annual coupons at acoupon rate of 5.6 percent and has a parvalue of $1,000. If these bonds have a

market price of $1,035, what yield tomaturity is being implied in the pricing?

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Using a financial calculator

[TI-BAII+] 4 x 2 = 8 [N]

1035 [PV] current market price

1000 x 0.056 2 = 28[+/-] [PMT] 1000 [+/-] [FV]

CPT

[I/Y] 2.3157

The expectedannualYTM is 2.3157x 2 = 4.63%


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Additional Problems 3:Zero Coupon Bond

Krypton Inc. wants to raise $3 million byissuing 10-yearzero couponbonds with aface value of $1,000. Their investment bankerinforms them that investors would use a9.25% discount rate on such bonds.

[3A] At what price would these bonds sell inthe market place assuming semi-annual

compounding?

[3B] How many bonds would the firm have toissue to raise $3 million?

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Additional Problems 4:Tax on zero-coupon bond income

Lets say that you buy 100 of the 7411 bondsthat were issued by Krypton Inc. asdescribed in Problem 3 above for $404.85.At the end of the year, how much moneywill the bond be worth, and how much taxwill you be assessed assuming that you havea marginal tax rate of35%?

Additi l P bl 4 (A )

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[1] Get the current bond price

[2] Find the bond price after one year

[3] Find the implied interest = price differences = [2] [1]

[4] Compute the tax on the interest = 0.35 x [3]

[1] Current Price = 404.85

[2]The bond price after 1 year with YTM of 9.25%:

20 2 = 18 [N] 9.25 2 = 4.625 [I/Y] 0 [PMT] 1000 [+/-] [FV] CPT [PV] 443.16

[3] Implied interest earned = price differences = [2] [1]= Price after one year Current Price

= $443.16 - $404.85 = $38.31

[4] Taxes due =Tax rate * Taxable income

= 0.35 * $38.31 = $13.41

Additi l P bl 4 (A )

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[3]The bond price after 1 yearwith YTM of 9.25%:

19 1 = 18 [N] 9.25 2 = 4.625 [I/Y]

0 [PMT]

1000[+/-] [FV]

CPT [PV] 443.16

[3] Implied interest earned

= Price at 12 months Current Price

= $443.16 - $404.85 = $38.31

[4] Taxes due =Tax rate * Taxable income

= 0.35 * $38.31 = $13.41

Additional Problems with Answers

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Additional Problems with AnswersProblem 5

Price, and BEY, on a Treasury bill:Calculate the price, and BEY of a treasury billwhich matures in 181 days, has a face valueof $10,000, and is currently being quoted at a

bank discount yield of 2.32%.

Additional Problems with Answers

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Additional Problems with AnswersProblem 5 (Answer)

Price of T-bill= Face Value * [1-(discount yield * days until maturity/360)]

= $10,000 * [ 1 - (.0232 * 181/360)]

= $10,000*0.98833555

= $9,883.36

BEY= 365 * Bank discount yield = 365 * .0232

360 - (days to maturity * discount yield) 360 - (181*.0232)

= .023799 = 2.38% (rounded to 2 decimals)

Figure 6.3 Future cash flow of a

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Figure 6.3 Future cash flow of aMerrill Lynch bond

FIGURE 6.7 Goodyear

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FIGURE 6.7 Goodyearsemiannual corporate bond.

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TABLE 6.5 AnnualInterest Rates onCorporate Bonds RatedAaa to Baa, 1980 to 2006

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FIGURE 6.10 Pacific Bell callable

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bond cash flows.

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