Chapter 7 Bonds and Interest Rates
The valuation of bonds involves calculation of the present value of an expected annuity payments (coupons) and the par value at maturity. Bond yields, both promised (yield to maturity) and realized (holding period return) are discussed, as is the inverse relationship between bond prices and interest rates.
1 BOND CHARACTERISTICS
A. A bond is a long-term promissory note issued by a business or government, i.e., corporate bonds (issued by business), Treasury bonds
(issued by U.S. federal government), municipal bonds or "munis" (issued by state or local government), and foreign bonds (issued by foreign governments or foreign corporations).
B. A bond promises to pay periodic interest or coupon to the bondholder at the coupon rate , plus return the face value principal amount borrowed at maturity.
C. For a fixed coupon rate bond, the coupon rate stays the same throughout the life of the bond. The interest rate is the market rate, or discount rate, at the time the bond is issued. Thereafter, the market rate may vary; the coupon rate, which determines the periodic coupon payment, stays the same.
D. Bonds generally have a specified maturity date on which the par value must be repaid. Original maturity is the number of years to maturity at the time a bond is issued. Effective maturity : the number of years to maturity from now on.
Example:
In 1995 U.S. government issued a 30-year T-B. "30" years is the original maturity. In 1998, the effective maturity of the 30-year T-B is "27" years.
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Reading the Financial Pages
A. U.S. Treasury and Agency securities prices are quoted in 32nds on a bid
(dealer offer to buy) and ask (dealer offer to sell) basis. The yield to maturity (based on asking price) is also quoted.
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B. Corporate bond prices are quoted in eighths with the company name, coupon rate, maturity date, current yield, volume, closing price, and daily price change, respectively.
BOND PRICES AND YIELDS
Bond Pricing
Annual coupon bonds:
PV = CP / (1+r)1 + CP / (1+r)2 +…+ CP / (1+r)N + FV/ (1+r)N
N
N
I/Y r
PV
?
PMT FV
CP 1000
Semiannual coupon bonds:
PV = 0.5CP / (1+r/2)1 + 0.5CP / (1+r/2)2 +…+ 0.5CP / (1+r/2)2N + FV/ (1+r/2)2N
N I/Y PV PMT FV
2N r/2 ? CP/2 1000
PV is the bond's value.
CP is the coupon payment. r is the bond's annual market rate of interest; it is the discount rate used to calculate PV. It is also called required (or expected) rate of return.
N is the effective maturity (number of years to maturity).
FV is the par (face) value (= $1,000).
Example:
Suppose you want to buy a 10-year bond that has face value of $1000 and annual coupon interest rate of 12%. The coupon payment is made every 6 months. The appropriate interest rate on the bond is 20%. What is the value of the bond? (Draw a time line).
CP = .12 x 1000/2 = $ 60, r = .20/2 = 10%, N = 10 x 2=20 periods, FV = $1000
PV = 60 / (1+10%)1 + 60 / (1+10%)2 +…+ 60 / (1+10%)20 + 1000 / (1+10%)20
= 510.81+148.64 = $ 659.45
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* From the bond valuation model, we know: a.

coupon interest rate,

PV of a bond b .

r,

PV and vice versa. c.
If coupon rate > r, then PV > Par Value = 1000 (a premium bond),
If coupon rate < r, then PV < Par Value = 1000 (a discount bond),
If coupon rate = r, then PV = Par Value = 1000 (a par bond).
The Yield to Maturity versus Current Yield
A. The current yield , calculated by dividing the annual coupon payment by the price of the bond, is a rough approximation of the expected return on the bond.
B. A better estimate of the expected return on the bond is the yield to maturity (YTM). The YTM is the interest rate for which the PV of the bond cash flows (coupons and face value) equals the bond price. The
YTM assumes that one will hold the bond until maturity.
C. A fixed rate bond means the coupon rate (which determines the amount of annual coupon payment) remains the same over the life of the bond.
Market interest rates (discount rates) change every day. Bond prices vary to give the new bond buyer the market rate of return.
Example:
Suppose a bond pays coupons once a year, and has $1,000 par value, 10% coupon rate, and maturity of 15 years. The discount rate is 10% today. The discount rate may be constant or change to either 5% or 15% in a year. The change of PV under different “r” is shown below.
Year r = 5% in 1 year
0 1,000.00
1
2
3
:
7
:
1,494.96
1,469.69
1,443.16
1,323.16
14 1,047.61
15 1,000.00 r is constant
1,000
1,000
1,000
1,000
1,000
1,000
1,000 r = 15% in 1 year
1,000.00
713.78
720.84
728.97
775.63
956.52
1,000.00
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* When r decreases to 5% which is below the coupon rate, the bond is sold at premium.
However, as year to maturity becomes smaller, the "premium" decreases as the year to maturity decreases. Finally, when maturity is zero, the "premium" decreases to zero and
PV =Par Value.
* When r increases to 15% which is above the coupon rate, the bond is sold at discount.
However, as year to maturity becomes smaller, the "discount" decreases as the year to maturity decreases. Finally, when maturity is zero, the "discount" decreases to zero and
PV =Par Value.
Example:
A newly issued 20-year bond with face value $1,000 and coupon rate 10% pays coupon payment every six months . The annual required rate of return is 10% for this bond and will stay constant throughout the life of the bond. What is the value of the bond,
1. when it is just issued?
2. when it becomes a 10-year bond?
$1,000
$1,000
3. when it becomes a 5-year bond? $1,000
4. when it is at maturity date? $1,000
5. In each of the above 4 cases, what’s the current yield?
* When the coupon rate = required rate of return, no matter what the maturity is and how often the coupon is paid, the PV = Par Value.
Example:
A newly issued 20-year bond with face value $1,000 and coupon rate 10% pays coupon payment every six months . The annual required rate of return is 10% for this bond today and will decrease to 5% in 1 year, then keeps constant throughout the life of the bond. What is the value of the bond,
1. when it is just issued? $1,000
2. when it becomes a 10-year bond? $1,389.73
3. when it becomes a 5-year bond? $1,218.80
4. when it is at maturity date? $1,000
5. In each of the above 4 cases, what’s the current yield?
Example:
A newly issued 20-year bond with face value $1,000 and coupon rate 10% pays coupon payment every six months . The annual required rate of return is 10% for this bond today and will increase to 15% in 1 year, then keeps constant throughout the life of the bond. What is the value of the bond,
1. when it is just issued? $1,000
2. when it becomes a 10-year bond?
3. when it becomes a 5-year bond?
$745.13
$828.39
4. when it is at maturity date? $1,000
5. In each of the above 4 cases, what’s the current yield?
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Interest Rate Risk
A. Bond prices (PV) vary inversely with changes in market interest rates.
B. The longer (shorter) the maturity of the bond, the greater (less) the change in the bond price for every change in bond discount rates.
C. If the bond is sold before maturity, the seller will receive the market price which, depending on the current market rate since the bond was issued, can be higher or lower than the face value.
D. The actual rate of return earned on the bond investment or the realized holding period return on the bond may be higher or lower than the YTM.
Example : price changes of a 30-yr bond vs. a 3-yr bond.
Default Risk and Bond Ratings
A. The market yield to maturity on bonds, other than U.S. Treasury bonds, includes a default or credit risk premium , or added yield to cover the market’s expected default loss on risky bonds.
B. The credit risk premium is the difference between the yield on a risky bond and a U.S. Treasury bond of similar maturity.
C. The higher the expected loss in yield from the risky bond, the higher the credit risk premium.
D. Bond rating firms, like Moody’s and Standard and Poor’s, rate the default risk of risky bonds.
E. High investment grades are in the range from AAA to BBB; speculative or junk bonds are rated BB or below.
F. Relative credit risk premium is measured by the vertical yield differences between securities of varying default risks.
Bond Markets
Corporate bonds are traded primarily in the OTC market. Because most bonds are owned by and traded among the large financial institutions, it is relatively easy for the OTC bond dealers to arrange the transfer of large blocks of bonds among the relatively few holders of the bonds.
Yield Curve
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