i
)
2
+
…
+
C
n
+
F
n
(
1
+
i
)
n
Discount bond, yield>coupon
Par, Premium, and Discount Bonds
•
Par bond
•
If a bond’s coupon rate is
equal to
its yield, the price will
equal the face value
•
Discount bond
•
If a bond’s coupon rate is
less than
its yield, the price will b
e less than
the face
value
•
Likely to happen
when interest rates are rising
•
Premium bond
•
If a bond’s coupon rate is
more than
its yield, the price will be
higher than
the
face value
•
Likely to happen when
interest rates are falling
Semiannual Compounding
•
Most bonds issued in Europe pay annual coupons, but bonds issued in the
U.S.
usually pay semi-annual coupons
Equation 8.2
P
B
=
C
m
1
+
i
m
+
C
m
(
1
+
i
m
)
2
+
C
m
(
1
+
i
m
)
3
+
…
+
C
m
+
F
mn
(
1
+
i
m
)
mn
Calculator Example – Bond Valuation
What is the price of a three-year, 5% coupon bond with a market yield of 8% and semi-
annual coupon payments?
Semi-annual yield = 8%/2=4%
Semi-annual coupon payment = $50/2=$25
Number of periods = 3x2=6
Zero Coupon Bonds

•
Zero Coupon bonds pay
face value
at maturity, but
do not make coupon payments
•
The yield of a zero coupon bond is therefore the
present value of the par value
•
Zero coupon bonds
pay cash only at maturity
; they sell for less than similar bonds
which make periodic interest payments
Equation 8.3
P
Z
=
F
mn
(
1
+
i
m
)
mn
What is the price of a zero coupon bond with a $1,000 face value, 10-year maturity, and semi-
annual compounding?
The market rate on similar bonds is 12%.
P
Z
=
$
1000
(
1
+
0.12
2
)
20
=
$
1000
(
1
+
0.06
)
20
=
$
311.80
Yield to Maturity
到期收益率
•
Yield to Maturity (YTM)
is the rate that makes the
present value of the bond’s cash
flows equal to the price of the bond
•
i.e., YTM is the rate a bondholder earns if the bond is held to maturity and all
coupon and principal payments are made as promised
•
YTM changes daily as interest rates change
•
Yield goes down, the price goes down
•
Long term bonds are more sensitive than small bonds
Effective Annual Yield
•
In bond trading, the effective annual rate (EAR) is called the effective annual yield
(EAY)
EAY
=
(
1
+
quoted rate
m
)
m
−
1
•
Simple annual yield is yield per period multiplied by the number of compounding
periods
; for bonds with annual compounding, simple annual yield is two times the
semi-annual yield
Calculator Example – YTM and EAY
An investor buys a 30-year bond with a $1,000 face value for $800. The bond’s coupon rate
is 8% and interest payments are made semi-annually. What are the bond’s yield to maturity
and effective annual yield?
Bond Yields
•
Realized Yield is the return earned on a bond given the cash flows actually received
by the investor
实际收益率是投资者实际收到的现金流量所产生的收益
•
It represents the interest rate at which the
present value of actual cash flows
generated by the investment equals the bond’s price
•
The realized yield is important because
it allows investors to see what they actually
earned on their investments
Interest Rate Risk
•
Bond theorems are statements about the math used in bond pricing
•