ch7

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Class Notes: Set 3 Valuing Bonds

BA 3341 Business Finance

Nataliya Polkovnichenko, Senior Lecturer

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Outline• Bond Definitions• Valuation of Bonds• Yield to Maturity (YTM)• Rate of Return• Interest Rate Risk• Default Risk

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What is a BOND?

• A Bond is an agreement ( or financial asset ) in which an issuer is required to repay to the investor the amount borrowed plus interest over a specified period of time.

Principal borrowed by government(corporation) from investor

Principal returned to investor

Principal Regular interest

payments

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Example In the beginning of 1994 U.S. government borrowed

$100 million at 9% yearly rate by issuing 100,000 bonds for $1,000 each. Each bond promised to pay its holder $90 (9% of $1,000) at the end of year for 10 years and repay $1,000 at the end of 2003.

$90 $90 $90

Years 1994 1995 2002 2003 …

You pay price of the bond

coupons

principal+last coupon

$1,000+$90=$1,090

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FOUR COMPONENTS OF A BOND

• Issuer The organization responsible for ensuring that interest and principal payments are made to bondholder.

• Principal The amount of money denominated in a specific currency that the issuer wishes to borrow and agrees to repay the investor. (Face Value, Par value, Maturity value)

• Coupon (rate) The rate of interest the issuer agrees to pay the investor. Usually stated as a percentage of the Face V.

• Maturity date The date on which the issuer of a bond must repay the principal due and the final interest rate payment.

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Bond Valuation ???Pricing a bond is a simple two step process:

1. Define the cash flows.

2. Calculate the aggregate present value of the cash flows using appropriate discount rate (r).

Note: Generally discount rate IS NOT EQUAL to

coupon rate!!!

c c c

Years 1 2 t-1 t …

You pay price of the bond

coupons

principal+last coupon

FaceValue+c

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What is appropriate discount rate (r)?

• It is an interest rate on similar bonds on financial markets.

• What does similar mean?• Similar features!!! (ALL)• Similar risk!!!

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Yield to Maturity (YTM)

• Yield to Maturity (YTM) - The single discount rate (internal rate of return) used to calculate the market price of the bond.

– Reflects the required market interest rate for the bond.

– Assumes the bond is held to maturity.– Different than the coupon rate.

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Yield to Maturity (YTM)

• In other words:• If you plan to hold the bond until maturity n years from

today:• The yield to maturity is the discount rate at which the

bond’s price is the present value of the bond’s payments (coupons and face value).

• YTM is the required return on the bond assuming it will be held until maturity.

nr

ParC

r

C

r

C

r

CbondPVice

1

...111

= )(Pr 32

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Bond Valuation (using YTM)

c c c

Years 1 2 t-1 t …

You pay price of the bond

coupons

principal+last coupon

FaceValue+ c

tr

FaceVC

r

C

r

CBondPV

)1(

....

)1()1()(

21

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Example Five year 9% coupon bond offers these cashflows.

Discount rate (YTM) is 9% .

What is the present value of this bond?

$90 $90 $90

Years 1999 2000 2002 2003

$1,000+$90=$1,090

$90

20011998

54321 )1(

.

)1()1()1()1()(

r

FaceVC

r

C

r

C

r

C

r

CBondPV

000,1)09.1(

090,1

)09.1(

90

)09.1(

90

)09.1(

90

)09.1(

90)(

54321

BondPV

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What happens if the discount rate (YTM) drops to 5.3%?

87.158,1)053.1(

090,1

)053.1(

90

)053.1(

90

)053.1(

90

)053.1(

90)(

54321

BondPV

If the interest rate on similar bonds was 5.3%the value of 9% Treasures increases to $1,158.87.

What happens if the interest rate on similar bonds was 10%?

09.962)1.1(

090,1

)1.1(

90

)1.1(

90

)1.1(

90

)1.1(

90)(

54321

BondPV

The value of 9% Treasures decreases to $962.09.

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Bond ValuationUsing the Annuity Formula

• Value a Bond with a combination of the Annuity and the Present Value formulas.

nn r+1

AmountPar

r+1r

1

r

1Coupon = Bond of Value

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Example• Suppose investors demand an interest rate of 5 percent on

a 10-year 6 percent bond. How much should investors pay to acquire the bond? (Alternatively, what’s the price of the bond?)

• Solution: coupon rate is 6%, this means we are going to receive $60 each year for 10 years and another $1,000 in the end of year 10. The discount rate is 5%.

22.077,1$05.1

1000

05.1*05.

1

05.

160

r+1

par

r+1r

1

r

1cpn PV

1010

tt

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Semiannual Coupon Bond

• In the United States, most bonds make coupon payments semiannually.

• A bond has a coupon rate of 6 percent, with coupon payment made semiannually. This means the bond makes payment of $1,000*(6%/2)=$30 every 6 months.

• Accordingly, the quoted interest rate is APR and has to be compounded semiannually.

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Semiannual Coupon Bond What is the value of a bond with a face value of

$1000, coupon rate of 10% and maturity of 10 years? Assume semiannual coupon payments and an annual interest rate of 8% compounded semiannually.

1136$)04.01(

)000,150(....

)04.01(

50

)04.01(

50Price

20210

0 1 2 3 4 5 20

$1000$50 $50$50$50$50 $50

Semiannual Coupon payments=(1000*0.10)/2

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Semiannual Coupon Bond

tbondcoupSem r

FaceVC

r

C

r

CPV

221.. )2/1(

....

)2/1()2/1(

2

.*. RateCouponValueFaceC

Semiannual payment :

Present Value of Semiannual Coupon Bond:

Where: r is annual discount rate (YTM)

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Using Financial Functions

FV=Par

Yearly discount Rate = I/YR

PMT PMT PMT

Years 0 1 2 N

PV = Price

Given any four of the variables

(N, I, PV, PMT,FV) your

financial calculator can solve

for the fifth.

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Examples

• $1,000 bond for 10 years with 9% interest and 9% yearly coupons

• PMT=90 FV=1000 N=10 I/Y=9% PV=?• $1,000 bond for 5 years with 5% interest and 9%

semiannual coupons• PMT=45 FV=1000 N=5*2 I/Y=5/2 PV=?• $1,000 bond for 10 years with 9% interest and 9%

yearly coupons .What happens after five years if the interest rate changes to 5% then? 11%?

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How can we calculate YTM?

• Find r so present value of payments is equal to price.

• Use financial calculator.• Example: $1,000 face value, $90 annual

coupons for 5 years, $1,100 price• FV=1000 PMT=90 N=5 PV=-1100 I/Y=?

Note: PV is equal to price!!!

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More Examples

Example 1: C = $90, t = 10 years, Po = 1050.FV=1000 PMT=90 N=10 PV=-1050

YTM=8.25%

Example 2: C = $90, t = 10 years, Po = 900.FV=1000 PMT=90 N=10 PV=-900

YTM=10.67%

%57.81050$

90$

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How do Bond Prices Vary with Interest?

• What happens if the coupon rate is equal to the interest rate? (see slide 11)

• What happens to bond prices if the interest rate goes up? (see slide 12)

• What happens to bond prices if the interest rate goes down? (see slide 12)

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Bond Prices and Interest Rates (YTM)

Int.rate=Coupon rate

Bond Price=Face Value Interest Rate

Bon

d P

rice

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Conclusion/Observation:

–When the market interest rate equals the coupon rate, bonds sell for the face value.

–When the market interest rate exceeds the coupon rate, bonds sell for less than face value.

–When the market interest rate is below the coupon rate, bonds sell for more than face value.

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Price vs. Face ValueIf P < Face V. the bond is said to be sold at a discount

If P > Face V. the bond is said to be sold at a premium

If P = Face V. the bond is said to be sold at par

Ex 1: If a bond with a face value FV = 100 would be sold in the market for P = 99.5 we would say it is sold at a discount, as in this case the price is smaller than the face value.

Ex 2: Suppose you are told that a bond with a face value of $50 is sold at par. What can you say about the price of this bond?

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Interest Rates or Yields Up (Down) = Prices Down (Up)

• As interest rates increase, the value of a bond decreases; conversely as interest rates decrease, the value of a bond increases. Therefore generally the following holds:

• coupon rate < required yield, then price < face value

bond is said to be selling at a discount. • coupon rate = required yield, then price = face value

bond is said to be selling at par. • coupon rate > required yield, then price > face value

bond is said to be selling at a premium.

MEMORIZE!!!

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Bond Prices Over Time

BondPrice

time

1 2 3 …

$1000

Premium Bond

Discount Bond

Maturity

?

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Rate of Return

We define Rate of Return over a specified period of time as:

The price change divided by the bond price today is called Capital Gain (or Loss).

todayPrice

change price+incomeCoupon =return of Rate

investment

income total=return of Rate

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Rate of Return

• Expected vs. Realized return.

• P0 – price paid for a bond,

• P1 – expected bond price at time 1 , when you sell the bond,

• C – coupon payments received between buying and selling

0

01 )( Returnof Rate

P

PPC

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Example 1:You pay $900 for a bond which has an annual coupon of $120. You sell it for $930 after one year. What is your rate of return?

$900

$900)-($930+$120= return of Rate

Rate of return=16.67%

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More Definitions• Current Yield - The annual coupon of the bond

divided by price.

• This return ignores the effect of price changes (capital gain or loss).

• The current yield is a rough approximation of the expected return on the bond.

Price

Coupon AnnuleldCurrent Yi

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Example 1: C = $90, t = 10 years, Po = 1050.

Current yield =

Example 2: C = $90, t = 10 years, Po = 900.

Current yield =

%57.81050$

90$

%10900$

90$

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Rate of Return versus YTM

Is the rate of return higher than or lower

than the YTM?

What can we say if the YTM does not change?

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Rate of Return versus YTM• Example 1: John buys a bond when its time to maturity

is 9 years, r is 12% and sells after a year when r is 10%. The coupon rate is 8%. Find the rate of return from this investment.

• Example 2: A bond pays annual coupon of $100. Ann bought the bond 1 year back when YTM was 9%. If Ann sells the bond today, YTM now being 9.5%, find the annual rate of return on her bond. This bond will mature in another 5 years.

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Rate of Return versus YTM

The above examples illustrate:

1. If YTM decreases (ex.1), the rate of return is higher then the initial YTM. Why? (Price of the bond grows)

2. If YTM increases (ex.2), the rate of return is lower then the initial YTM. Why? (Price of the bond goes down)

3. If YTM does not change_____________________.

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Interest Rate Risk

• Bond price fluctuates as interest rate changes.

• If interest rate suddenly goes up, will your bond be worth more or less? What if interest rate goes down?

• Will a 3-year bond or a 30-year bond be subject to more interest rate risk?

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Interest Rate Risk

• Two bonds: a 6% coupon bond with 3 years to maturity and a 6% coupon bond with 30 years to maturity. Both sell at face value today ($1,000). What is the discount rate?

• What would be the price of each bond today if the discount rate

• drops to 4%?

• jumps to 8%?

PV3=1055.5 ;% change=5.6%

PV30 =1345.84 ; % change=34.6%

PV30=774.84 ; % change= -22.5%

PV3 =948.46 ; % change= -5.1%

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Interest Rate Risk

-

500

1,000

1,500

2,000

2,500

3,000

Discount Rate

$ B

on

d P

ric

e

30 yr bond

3 yr bond

When the discount rate(yield) equals the 6% coupon rate, both bonds

sell at face value

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Interest Rate Risk

• The fluctuations in price due to changes in interest rates is called interest rate risk. This risk is presented in a bond even if the future cash flows are risk free or will be received with certainty.

• This risk increases as maturity increases. In another word, long term bond prices are more sensitive to the interest rate than prices of short-term bonds.

• Interest rate risk increases as coupon rate decreases.

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Yield Curve / Term Structure• The relationship between yields and time to maturity is

called a yield curve

• The Yield curve for Treasury Bonds is in the Wall Street Journal every day.

• These yields are influenced by the expectations of future interest rates and inflation.

2%

3%

4%

5%

6%

7%

5Yr 10Yr 15Yr 20Yr 25Yr 30Yr

Yield

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Default Risk• Coupons and principal will always be repaid if

U.S. government is the borrower.• What about corporations?• Because there is a default risk, there is a default

premium.• Default premium is the difference between yield

on corporate bond and on U.S. government bond of same coupon and maturity.

• The two largest rating agencies are Moody’s and Standard & Poor’s.

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RatingsMoody’s S&P

• Aaa• Aa• A• Baa• Ba• B• Caa• Ca• C

• AAA• AA• A investment grade• BBB bonds

• BB “junk “bonds or• B speculative grade • CCC• Cc• C

• D bonds in default

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Example

• A bond’s credit rating provides a guide to its risk. Long-term bonds rated Aa currently offer yield to maturity of 7.5 percent. A-rated bonds sell at yield of 7.8 percent. If a 10-year bond with a coupon rate of 7 percent is downgraded by Moody’s from Aa to A rating, what is the likely effect on the bond price?

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Questions

• Are all bonds subject to default risk?

• Are all bonds subject to interest rate risk?

• Is a default-free bond risk-free?

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Other Types of Bonds• Floating Rate Bonds - Bonds where the coupon rate

can change over time.• Convertible Bonds - Bonds where a right (option) is

granted to the investor to exchange the bond for some other specified security, usually common stock, before the Maturity Date.

• Zero-Coupon Bonds - Bonds that pay only the face value at maturity, no coupons.

• Callable Bonds - Can be bought back early.• Put Bond -allows holder to redeem for par.

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Zero-Coupon Bond Pricing

Price Maturity(Years)

YTM

$300 30

$300 8%

10 10%

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Government Bonds

• Treasury bills:bonds with maturity of one year or less

• Treasury notes:bonds with maturity from 1 year to 10 years

• Treasury bonds:maturity more than 10 years

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What you should know about bonds

• Distinguish among the bond’s coupon rate, current yield, and yield to maturity.

• Find the market price of a bond given its yield to maturity, find a bond’s yield given its price, and demonstrate why prices and yields vary inversely.

• Show why bonds exhibit interest rate risk.

• Understand why investors pay attention to bond ratings and demand a higher interest rate for bonds with low ratings.