Chapter 11: Bond Prices and Yields
11.1 Bond Basics
Straight bond - is an IOU that obligates the issuer of the bond to pay the holder of the bond:
A fixed sum of money (called the principal, par value, or face value) at the bond’s maturity and sometimes
Constant, periodic interest payments (called coupons) during the life of the bond.
Treasury bonds are straight bonds.
Special features may be attached:
o Convertible bonds
o Callable bonds
o Putable bonds
Coupon Rate and Current Yield
Two basic yield measures for a bond are its coupon rate and its current yield.
Coupon Rate – a bond’s annual coupon divided by its par value. Also called coupon yield or nominal
Coupon rate= Annual coupon
Current yield – a bond’s annual coupon divided by its market price
11.2 Straight Bonds and Yield to Maturity
Yield to Maturity (YTM) – is the discount rate that equates today’s bond price with the present value of the future
cash flows of the bond.
o Sometimes called its promised yield
The price of a bond is found by adding together the present value of the bond’s coupon payments and the present
value of the bond’s face value.
Straight Bond Prices
C 1 FV
Bond Price= 1− 2M + 2M
YTM 1+ YTM 1+ YTM
( 2) ( 2)
present value of the bond’s coupon payments
present value of the bond’s face value
C represents the annual coupon payments (in $)
FV is the face value of the bond (in $)
M is the maturity of the bond, measured in years
YTM is the yield to maturity
What is the price of a straight bond with: $1,000 face value, coupon rate of 8%, YTM of 7%, and a maturity of 20
C 1 FV
Bond PriceYTM − 1+YTM 2M+ 1+YTM 2M
( 2) ( 2)
Bond Price =0 − 1 2×2+ 10002×20
0.07 (1+0.072) (1+0.072)
= (1,142.857×0.747428) + 252.5725
Premium and Discount Bonds
Bonds are given names according to the relationship between the bond’s selling price and its par value.
Premium bonds price> par value YTM < coupon
rate Discount bonds price< par valueYTM > coupon
Par bonds price = par YTM = coupon
In general, when the coupon rate and YTM are held constant:
o For premium bonds: the longer the term to maturity, the greater the premium over par value.
o For discount bonds: the longer the term to maturity, the greater the discount from par value.
Relationships Among Yield Measures
For premium bonds:
o coupon rate > current yield > YTM
For discount bonds:
o coupon rate < current yield < YTM
For par value bonds:
o coupon rate = current yield = YTM
Suppose we know the current price of a bond, its coupon rate, and its time to maturity. How do we calculate the
We can use the straight bond formula, trying different yields until we come across the one that produces the
current price of the bond.
$110= $80 1− 1 + $1,000
YTM YTM 2× YTM 2×8 YTM = 6.3843%
(1+ 2) (1+ 2)
A Note on Bond Price Quotes
If you buy a bond between coupon dates, you will receive the next coupon payment (and might have to pay taxes
However, when you buy the bond between coupon payments, you must compensate the seller for any accrued
The convention in bond price quotes is to ignore accrued interest.
o Clean Price – the price of a bond net of accrued interest; this is the price that is typically quoted
o This results in what is commonly called a clean price (i.e., a quoted price net of accrued interest).
o Sometimes, this price is also known as a flat price.
The price the buyer actually pays is called the dirty price.
o Dirty Price – the price of a bond including accrued interest, also known as the full or invoice price
o This is because accrued interest is added to the clean price.
o Note: The price the buyer actually pays is sometimes known as the full price, or invoice price.
11.3 More on Yields
Yield To Call
Most bonds are callable bonds –a bond is callable if the issuer can buy it back before it matures
A callable bond gives the issuer the option to buy back the bond at a specified call price anytime after an initial
call protection period.
o Call price – the price the issuer of a callable bond must pay to buy it back
o Call protection period (call deferment period) – the period during which a callable bond cannot be
Therefore, for callable bonds, YTM may not be useful.
Yield to call (YTC)– is a yield measure that assumes a bond will be called at its earliest possible call date.
The formula to price a callable bond is:
C 1 CP
Callable Bond Price= 1− 2T+ 2T
YTC (1+ YTC ) (1+ YTC )
C is the annual coupon (in $)
CP is the call price of the bond
T is the time (in years) to the earliest possible call date YTC is the yield to call, with semi-annual coupons.
o As with straight bonds, we can solve for the YTC, if we know the price of a callable bond.
11.4 Interest Rate Risk and Malkiel’s Theorems
Holders of bonds face interest rate risk.
Interest rate risk - is the possibility that changes in interest rates will result in losses in the bond’s value.
Realized Yield - the yield actually earned or “realized” on a bond
Realized yield is almost never exactly equal to the yield to maturity, or promised yield.
1. Bond prices and bond yields move in opposite directions.
a. As a bond’s yield increases, its price decreases.
b. Conversely, as a bond’s yield decreases, its price increases.
2. For a given change in a bond’s YTM, the longer the term to maturity of the bond, the greater the magnitude of the
change in the bond’s price.
3. For a given change in a bond’s YTM, the size of the change in the bond’s p