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 Basics
When 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.
6-2
Bond Pricing: Example
Treasury 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.
6-3
Bond Pricing
coupon coupon
(coupon  par )
PV 

 .... 
1
2
t
(1  r )
(1  r )
(1  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.
6-4
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%.
6-5
Bond Pricing
A bond is a package of two investments: an annuity
and a final repayment.
PVBond  PVCoupons  PVParValue
PVBond  coupon  ( Annuity Factor )  par value  ( Discount Factor )
1  (1  r )  t
where Annuity Factor 
r
1
and Discount Factor 
(1  r )t
6-6
Bond Pricing: Example
What 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%.
PVBond
1  (1  .04)3
1
 $90 
 $1, 000 
.04
(1  .04)3
 $1,138.75
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Bond Prices & Interest Rates
As 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%?
6-8
Bond Yields
To 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.
6-9
Current Yield: Example
Suppose 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.
6-10
Yield to Maturity
Yield to Maturity:
coupon coupon
(coupon par)
PV 

 ....
1
2
(1  r )
(1  r )
(1  r )t
• 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: Example
Suppose 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?
$60
$60
($60  $1,000)
$1,150 


1
2
(1  r )
(1  r )
(1  r ) 3
6-12
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.
6-13
Rate of Return: Example
Suppose 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?
6-14
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 & Inflation
In the presence of inflation, an investor’s real interest
rate is always less than the nominal interest rate.
6-16
Interest Rates & Inflation
If 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 Default
When 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.
6-18
The Risk of Default
Bonds 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.
6-20
Appendix A: Treasury Bond
Rates
10-year U.S. Treasury bond interest rates, 1900-2010
6-21
Appendix B: Real vs. Nominal Yields
Red line – Real yield on long-term UK indexed bonds
Blue line – Nominal yield on long-term UK bonds
6-22
Appendix C: Credit Ratings
6-23