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Lecture 6: Valuing Bonds

A bond is a debt instrument issued by governments or corporations to raise money

The successful investor must be able to:• Understand bond structure

• Calculate bond rates of return

• Understand interest rate risk

• Differentiate between real and nominal returns

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Bond BasicsWhen governments or companies issue bonds, they promise to make a

series of interest payments and then repay the debt.Bond

• Security that obligates the issuer to make specified payments to the bondholder.

Face Value• Payment at the maturity of the bond.• Also called “principal ” or “par value ”

Coupon• The interest payments paid to the bondholder.

Coupon Rate• Annual interest payment as a percentage of face value.

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Bond Pricing: ExampleTreasury bond prices are quoted in 32nds

rather than in decimals.

Example:For a $1000 face value bond with a bid price of 103:05 and an asked price of 103:06, how much would an investor pay

for the bond?

103% + (06/32) = 103.1875% of face value(1.031875) * ($1,000) = $1,031.875

• Asked Price – The price that investors need to pay to buy the bond.• Bid Price – The price asked by an investor who owns the bond and wishes to sell it.• Spread – The difference between the bid price and the asked price.

• The spread is how a seller of a bond makes a profit.

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Bond Pricing

1 2

( )....

(1 ) (1 ) (1 )t

coupon coupon coupon parPV

r r r

• The value of a bond is the present value of all cash flows generated by the bond (coupons and repayment of face value), discounted at the required rate of return.

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Bond Pricing: Example

What is the price of a 9% annual coupon bond with a par value of $1,000 that matures in 3 years? Assume a required

rate of return of 4%.

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Bond PricingA bond is a package of two investments: an annuity

and a final repayment.

( ) ( )

1 (1 )where

1and

(1 )

Bond Coupons ParValue

Bond

t

t

PV PV PV

PV coupon Annuity Factor par value Discount Factor

rAnnuity Factor

r

Discount Factorr

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Bond Pricing: ExampleWhat is the value of a 3-year annuity that pays $90 each year and an additional $1,000 at the date of the final repayment? Assume a discount rate of 4%.

3

3

1 (1 .04) 1$90 $1,000

.04 (1 .04)

$1,138.75

BondPV

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Bond Prices & Interest RatesAs interest rates change, so do bond prices.

What is the present value of a 4% coupon bond with face value $1,000 that matures in 3 years? Assume a discount rate of 5%.

What is the present value of this same bond at a discount rate of 2%?

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Bond YieldsTo calculate how much we earn on a bond investment, we can calculate

two types of bond yields:

Current Yield: Annual coupon payments divided by bond price.Yield to Maturity: Interest rate for which the present value of the bond’s payments equals the price.

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Current Yield: ExampleSuppose you spend $1,150 for a $1,000 face value bond that pays a $60 annual coupon payment for 3

years.

What is the bond’s current yield?

• Current Yield – Annual coupon payments divided by bond price.

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Yield to Maturity

Yield to Maturity:

tr

parcoupon

r

coupon

r

couponPV

)1(

)(....

)1()1( 21

• Yield to Maturity – Interest rate for which the present value of the bond’s payments equals the price.

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Yield to Maturity: ExampleSuppose you spend $1,150 for a $1,000 face value bond

that pays a $60 annual coupon payment for 3 years.

What is the bond’s yield to maturity?

321 )1(

)000,1$60($

)1(

60$

)1(

60$150,1$

rrr

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Rate of Return• A bond’s yield to maturity is only helpful if the investor plans on holding the bond until it matures.

• A bond’s rate of return can be calculated regardless of how long the bond is held.

• Rate of return – Total income per period per dollar invested.

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Rate of Return: ExampleSuppose you purchase a 5% coupon bond, par value $1,000,

with 5 years until maturity, for $975.00 today. After one year you sell the bond for $965.00.

What was the rate of return during the period?

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The Yield Curve: Example• Yield Curve – Plot of the relationship between bond yields to maturity and time to maturity.

• The yield curve usually slopes upwards, implying that long term bonds generally earn higher yields than short-term bonds.

• When interest rates are expected to rise, the yield curve is often upward sloping.

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Interest Rates & InflationIn the presence of inflation, an investor’s real interest

rate is always less than the nominal interest rate.

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Interest Rates & InflationIf you invest in a security that pays 10% interest annually and inflation is 6%, what is your real

interest rate?

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The Risk of DefaultWhen investing in bonds, there is always the

risk that the issuer may default.

•Default risk: The risk that a bond issuer may default on his bonds.• Companies compensate investors for bearing this added risk in the form of

higher interest rates on their bonds.

•Default premium: The additional yield on a bond that investors require for bearing credit risk.

• Usually the difference between the promised yield on a corporate bond and the yield on a U.S. Treasury bond with the same coupon and maturity.

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The Risk of DefaultBonds come in many categories, with returns commensurate with risk.

•Credit agency: An agency that rates the safety of most corporate bonds.• Examples: Moody’s, Standard & Poor‘

Investment-grade bonds: Bonds rated Baa or above by Moody’s or BBB or above by Standard & Poor’s.

Junk bonds: Bond with a rating below Baa or BBB

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Types of Corporate Bonds•Zero-Coupon Bonds – Bonds that are issued well below face value with no coupon payment. At maturity investors receive $1,000 face value for the bond.

• Are corporate bonds the only bonds which can be offered as zero-coupon bonds?

•Floating-Rate Bonds – Bonds with coupon payments that are tied to some measure of current market rates. A common example would be a bond with coupon rate tied to the short-term Treasury rate plus 2%.

•Convertible Bonds – Bonds that allow the holder to exchange the bond at a later date for a specified number of shares of common stock.

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Appendix A: Treasury Bond Rates

10-year U.S. Treasury bond interest rates, 1900-2010

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Appendix B: Real vs. Nominal Yields

Red line – Real yield on long-term UK indexed bondsBlue line – Nominal yield on long-term UK bonds

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Appendix C: Credit Ratings