Bonds and Bond Valuation

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1. Understand basic bond terminology and apply the time value of money equation in pricing bonds. 2.Understand the difference between annual and semiannual bonds and note the key features of zero-coupon bonds.3. Explain the relationship between the coupon rate and the yield to maturity. 4. Delineate bond ratings and why ratings affect bond prices.5. Appreciate bond history and understand the rights and obligations of buyers and sellers of bonds.6. Price government bonds, notes, and bills.

LEARNING OBJECTIVES

6.1 Application of the Time Value of Money

Tool: Bond PricingBonds --Long-term debt instruments Provide periodic interest income – annuity seriesReturn of the principal amount at maturity –

future lump sumPrices can be calculated by using present value

techniques i.e. discounting of future cash flows.Combination of present value of an annuity and

of a lump sum

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6.1 Key Components of a Bond

• Par value : Par value : Typically $1,000

• Coupon rate: Coupon rate: Annual rate of interest paid.

• Coupon:Coupon: Regular interest payment received by holder per year.

• Maturity date: Maturity date: Expiration date of bond when par value is paid back.

• Yield to maturity: Yield to maturity: Expected rate of return based on price of bond

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Table 6.1 Bond Information

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6.1 Key Components of a BondExample: Key components of a corporate bondLet’s say you see the following price quote

for a corporate bond:

 Issue Price Coupon(%) Maturity YTM% Current Yld. Rating

Hertz Corp. 91.50 6.35Hertz Corp. 91.50 6.35 15-Jun-2010 15.438 15-Jun-2010 15.438 6.94 6.94 B B

Price = 91.5% of $1,000.00 = $915.00$915.00

Annual coupon = 6.35% x $1,000.oo = $63.50$63.50

Maturity date = June 15, 2010June 15, 2010

If bought and held to maturity, yield (YTM) = 15.438%15.438%

Current Yield = Annual Coupon / Price = $63.50 / $915.00 = 6.94%6.94%

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6.1 Pricing a Bond in StepsSince bonds involve a combination of an annuity (coupons) and a lump sum (par value) its price is best calculated by using the following steps:

 

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6.1 Pricing a Bond in StepsExample: Calculating the price of a corporate bond.

Calculate the price of an AA-rated, 20-year, 8% coupon (paid annually) corporate bond (Par value = $1,000) which is expected to earn a yield to maturity of 10%.

Annual coupon = PMT = Coupon rate x Par value = .08 * $1,000 = $80$80YTM = r = 10%10%

Maturity = n = 2020Par Value = FV = $1,000.00$1,000.00

  Price of bond = Present Value of coupons + Present Value of par value

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Year 0 1

$80

2

$80

3

$80

20

$80$1,000

18 19

$80 $80…

6.1 Pricing a Bond in StepsExample: Calculating the price of a corporate bond 

Present value of coupons =

Present Value of Par Value =

Present Value of Coupons = $80 x 8.51359 = $681.09$681.09

Present Value of Par Value = $1,000 x 0.14864 = $148.64$148.64Price of bond = $681.09 + $148.64= $829.73$829.73

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rr1

11

PMTn

0.10

0.101

11

$8020

nr1

1FV

200.101

1$1,000

6.1 Pricing a Bond in StepsMethod 2. Using a financial calculator Mode: P/Y=1 C/Y = 1 (Because coupons are paid

annually)

 Key: N I/Y PV PMT FVInput: 20 10 ? 80 1000Compute --829.73829.73

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• Most corporate and government bonds pay coupons on a semiannual basis.

• Some companies pay no coupons, issuing 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 for number of coupon payments 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.

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6.2 Semiannual Bonds and Zero- Coupon Bonds

6.2 Semiannual Bonds

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6.2 Semiannual Bonds

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Using TVM Equation, YTM is 8.8%

6.2 Semiannual Bonds

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Using Financial Calculator, YTM is 8.8%

6.2 Semiannual Bonds

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6.2 Pricing Bonds after Original Issue

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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 until expiration.

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

6.2 Pricing Bonds after Original Issue  Example: Pricing a semi-annual coupon bond

after original issue:  Sixteen and 1/2 years after issue, price the Coca-

Cola bond issued as an 8.5% coupon (paid semi-annually), 30-year, A-rated bond at its par value of $1000. Currently, the yield to maturity on these bonds is 5.473%. Calculate the price of the bond today.

Remaining coupons, n = (60-33) = Remaining coupons, n = (60-33) = 2727Semi-annual coupon = (.085 x 1000)/2 = Semi-annual coupon = (.085 x 1000)/2 = $42.50$42.50Par value = Par value = $1,000.00$1,000.00Annual YTM = 5.473%, r = 5.473% / 2 = Annual YTM = 5.473%, r = 5.473% / 2 = 2.7365%2.7365%

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6.2 Pricing Bonds after Original Issue

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Method 1: Using TVM equations

Bond Price =

rr1

11

Couponr1

1 ValuePar

n

n

Bond Price =

0.0273650.0273651

11

$42.500.00273651

1$1,000

27

27

Bond Price = $1000 x 0.48243 + $42.50 x 18.91369 Bond Price = $482.43 + $803.83 Bond Price = $1,286.26

6.2 Pricing Bonds after Original Issue

Method 2: Using a financial calculator  

Mode: P/Y=2; C/Y = 2 

Key: N I/Y PV PMT FVInput: 27 5.473 ? 42.50 1,000Output --1,286.261,286.26

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6.2 Zero-Coupon Bonds

• Known as “pure” discount bonds and sold at a discount from face value

• Does not pay any interest over the life of the bond.

• At maturity, the investor receives the par value, usually $1000 which reflects the original purchase price (principal) and accumulated interest.

• Price of a zero-coupon bond is calculated by merely discounting its par value at the prevailing discount rate or yield to maturity.

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6.2 Amortization of a Zero-Coupon Bond

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•Interest earned is calculated for each 6-month period, first period is:

0.04 x 790.31 = $31.62$31.62•Interest is added to price to compute ending price,

$790.31 + $31.62 = $821.93$821.93•Zero-coupon bond investors have to pay tax on annual price appreciation even though no cash is received.

6.2 Amortization of a Zero-Coupon Bond

Example: Price of and taxes due on a zero-coupon bond:

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

  A) How much would he have to pay for it?

B) How much will he be taxed on the investment after 1 year, if his marginal tax rate is 30%?

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6.2 Amortization of a Zero-Coupon BondExample (Answer) First Price the Bond

Method 1: Using TVM equationBond Price = Par Value x [1/(1+r)n] Bond Price = $1000 x [1/(1.045)40]

  Bond Price = $1000 x 0.1719287 = $171.93$171.93

Method 2: Using a financial calculator   Mode:P/Y=2; C/Y = 2  Key: N I/Y PV PMT FV

Input: 40 9 ? 0 1000Compute --171.93171.93

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6.2 Amortization of a Zero-Coupon Bond

Example 4 (Answer—continued)Calculate the price of the bond at the end of 1 year.

Mode: P/Y=2; C/Y = 2Key: N I/Y PV PMT FVInput: 3838 9 ? 0 1000Compute --187.75187.75

 Taxable income = $187.75 - $171.93 = $15.82$15.82Taxes due = Tax rate * Taxable income = 0.30*$15.82 = $4.75$4.75

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6.3 Yields and Coupon Rates

• A Bond’s coupon rate differs from its yield to maturity (YTM).

• Coupon rate -- set by the company at the time of issue and is fixed (except for newer innovations which have variable coupon rates)

• YTM is dependent on market, economic, and company-specific factors.

• YTM varies across time as conditions or factors change.

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6.3 The First Interest Rate: Yield to Maturity

• Expected rate of return on a bond if held to maturity.

• The price that willing buyers and sellers settle at determines a bond’s YTM at any given point.

• Changes in economic conditions and risk factors will cause bond prices and their corresponding YTMs to change.

• YTM can be calculated by entering the coupon amount (PMT), price (PV), remaining number of coupons (n), and par value (FV) into the financial calculator or spreadsheet.

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6.3 The “Other” Interest Rate: Coupon Rate

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

• It represents the annual rate of interest that the 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 x $1,000 = $70.00 per year on each bond,

• It is usually paid either in a single check of $70.00 (annual) or two checks of $35.00 (semi-annual).

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6.3 Relationship of Yield to Maturity and Coupon Rate

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6.3 Relationship of Yield to Maturity and Coupon Rate

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6.3 Relationship of Yield to Maturity and Coupon Rate

Example: Computing YTM 

Last year, The ABC Corporation had issued 8% coupon (semi-annual), 20-year, AA-rated bonds (Par value = $1,000.00) to finance its business growth. If investors are currently offering $1,200.00 on each of these bonds, what is their expected yield to maturity on the investment? If you are willing to pay no more than $980.00 for this bond, what is your expected YTM?

 Remaining number of coupons = 19 x 2 = 38Semi-annual coupon amount =( .08 x $1,000)/2 = $40.00

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6.3 Relationship of Yield to Maturity and Coupon RatePV = $1,200.00Mode: P/Y=2; C/Y = 2Key: N I/Y PV PMT FVInput: 38 ? -1200 40 1000Compute 6.196.19 

Note: This is a premium bond, so it’s YTM of 6.19% < Coupon rate of 8%

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6.3 Relationship of Yield to Maturity and Coupon RatePV = $980.00Mode:P/Y=2; C/Y = 2Key: N I/Y PV PMT FVInput: 38 ? -980 40 1000Compute 8.218.21 

Note: This would be a discount bond so it’s YTM of 8.21% > Coupon rate of 8%

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6.4 Bond Ratings• Ratings are produced by Moody’s, Standard and Poor’s, 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 carry higher yields than those rated BBB or above (investment grade).

  Fallen angels: is the label given to bonds that have had their ratings lowered from investment to speculative grade. 6-33

6.4 Bond Ratings

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6.5 Some Bond History and More Bond Features• Corporate bond features have gone through some

major changes over the years.– Bearer bonds: – Indenture or deed of trust: – Collateral:– Mortgaged security: – Debentures: – Senior debt: – Sinking fund:– Protective covenants:– http://screen.yahoo.com/bonds.html

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6.5 Some Bond History and More Bond Features

Callable bond:Yield to call:Putable bond:Convertible bond:Floating-rate bond:Prime rate: Income bonds:Exotic bonds:

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6.5 Some Bond History and More Bond FeaturesExample: Calculating Yield to Call.  Two years ago, The Mid-Atlantic Corporation

issued a 10% coupon (paid semi-annually), 20-year maturity, bond with a 5-year deferred call feature and a call penalty of one coupon payment in addition to the par value ($1000) if exercised. If the current price on these bonds is $1,080, what is its yield to call?

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6.5 Some Bond History and More Bond Features

Remaining number of coupons until first call date, n = 6Semi-annual coupon = $50.00 = PMTCall price = $1,050 = FVBond price = $1,080 = PV

 Mode: P/Y=2; C/Y = 2Key: N I/Y PV PMT FVInput: 6 ? -1080 50 1050Compute 8.438.43

YTCYTC6-38

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 local

government agencies. • Treasury bills, are zero-coupon, pure discount

securities with maturities ranging from 1-, 3-, and 6-months up to 1 year.

• Treasury notes have between two to 10 year maturities.

• Treasury bonds have greater than 10-year maturities, when first issued.

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6.6 U.S. Government Bonds

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6.6 Pricing a U.S. Government Note or Bond

Similar to the method used for pricing corporate bonds and can be done by using TVM equations, a financial calculator or a spreadsheet program.

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

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6.6 Pricing a Treasury bill

Calculated by discounting the bill’s face value for the number of days until maturity and at the prevailing bank discount yield. Bank discount yield: is a special discount rate used in conjunction with treasury bills under a 360 day-per-year convention (commonly assumed by bankers).

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

 BEY = 365 x Bank discount yield

360 - (days to maturity x discount yield)

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6.6 Pricing a Treasury bill (continued)

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6.6 Pricing a Treasury billExample : Calculating the price and BEY of a Treasury bill.Calculate the price and BEY of a treasury bill which matures in 105 days, has a face value of $10,000 and is currently being quoted at a bank discount yield of 2.62%. Price of T-bill = Face value x [1-(discount yield * days until maturity/360)]

Price of T-bill = $10,000 x [ 1 - (.0262 x 105/360)] = $10,000 x 0.9923583

Price of T-bill = $9,923.58$9,923.58

BEY = 365 x Bank discount yield_________ = 365 x 0.0262 360 - (days to maturity x discount yield) 360 - (105 x 0.0262)

BEY = .026768 = 2.68%2.68% (rounded to 2 decimals)

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