Bond Prices: Relationship Between Coupon and Yield
• If YTM = coupon rate, then par value = bond price
• If YTM > coupon rate, then par value > bond price
▪ Why? The discount provides yield above coupon rate.
▪ Price below par value, called a discount bond
• If YTM < coupon rate, then par value < bond price
▪ Why? Higher coupon rate causes value above par.
▪ Price above par value, called a premium bond
There are the purely mechanical reasons for these results. We know that present values decrease as rates
increase. Therefore, if we increase our yield above the coupon, the present value (price) must decrease
below par. On the other hand, if we decrease our yield below the coupon, the present value (price) must
increase above par.
The Bond-Pricing Equation Adjusted for Semi-annual Coupons
C = Annual coupon payment C/2 = Semi-annual coupon
YTM = Annual YTM (as an APR) YTM/2 = Semi-annual YTM
t = Years to maturity 2t = Number of 6-month
periods to maturity
In practice, bonds issued in the United States usually make coupon payments twice a year
• If an ordinary bond has a coupon rate of
14 percent, then the owner will get a total
of $140 per year, but this $140 will come in
two payments of $70 each. The yield to
maturity is quoted at 16 percent. The bond
matures in seven years.
• How many coupon payments are there?
• What is the semiannual coupon payment?
• What is the semiannual yield?
• What is the bond price?
• Coupon rate = 14% – Semiannual
• YTM = 16% (APR)
• Maturity = 7 years – Number of coupon payments? (2t or N)
• 14 = 2 x 7 years
– Semiannual coupon payment? (C/2 or PMT)
• $70 = (14% x 1000)/2
– Semiannual yield? (YTM/2 or I/Y)
• 8% = 16%/2
Note: Bond yields are quoted like APRs; the quoted rate is equal to the actual rate per period multiplied by
the number of periods.
Coupon rate = 14%, semiannual coupons
YTM = 16%
Maturity = 7 years
Par value = $1,000
• Semiannual coupon = $70
• Semiannual yield = 8%
• Periods to maturity = 14
• Bond value =
• 70[1 – 1/(1.08)14] / .08 + 1000 / (1.08)14 = 917.56
( ) ( )2t
2 C Value Bond
Using the calculator:
CPT PV = -917.56
Interest Rate Risk
• Price Risk ▪ Change in price due to changes in interest rates
▪ Long-term bonds have more price risk than short-term bonds.
▪ Low coupon rate bonds have more price risk than high coupon rate bonds.
• Reinvestment Rate Risk ▪ Uncertainty concerning rates at which cash flows can be reinvested
▪ Short-term bonds have more reinvestment rate risk than long-term bonds.
▪ High coupon rate bonds have more reinvestment rate risk than low coupon rate bonds.
The risk that arises for bond owners from fluctuating interest rates is called interest rate risk. How much
interest rate risk a bond has depends on how sensitive its price is to interest rate changes. This sensitivity
directly depends on two things: the time to maturity and the coupon rate. As we will see momentarily, you
should keep the following in mind when looking at a bond:
All other things being equal, the longer the time to maturity, the greater the interest rate risk.
All other things being equal, the lower the coupon rate, the greater the interest rate risk.
If we compared a 10-year bond to a 1-year bond, we would see that the 10-year bond has much greater
interest rate risk. However, if you were to compare a 20-year bond to a 30-year bond, you would find that
the 30-year bond has somewhat greater interest rate risk because it has a longer maturity, but the difference
in the risk would be fairly small.
The value of a bond depends on the present value of its coupons. As a result, all other things being equal,
its value will fluctuate more as interest rates change. Put another way, the bond with the higher coupon has
a larger cash flow early in its life, so its value is less sensitive to changes in the discount rate.
Bonds are usually not issued with maturities longer than 30 years. However, low interest rates have led to
the issuance of bonds with much longer maturities. In the 1990s, Walt Disney issued “Sleeping Beauty”
bonds with a 100-year maturity. This company wanted to lock in the historically low interest rates for a
One potentially undesirable feature of high-coupon bonds is the required reinvestment of coupons at the
computed yield-to-maturity if one is to actually earn that yield. Those who purchased bonds in the early
1980s (when even high-grade corporate bonds had coupons over 11%) found, to their dismay, that interest
payments could not be reinvested at similar rates a few years later without taking greater risk. A good
example of the trade-off between interest rate risk and reinvestment risk is the purchase of a zero-coupon
bond – one eliminates reinvestment risk but maximizes interest-rate risk.
• Yield to Maturity (YTM) is the rate implied by the current bond price.
• Finding the YTM requires trial and error if you do not have a financial calculator and is similar to the process for finding r with an annuity.
• If you have a financial calculator, enter N, PV, PMT, and FV, remembering the sign convention (PMT and FV need to have the same sign, PV the opposite sign.)
Computing Yield to Maturity
YTM with Annual Coupons
Consider a bond with a 10% annual coupon rate, 15 years to maturity and a par value of $1000. The current price is $928.09. – Will the yield be more or less than 10%?
928.09 PV (enter as a negative)
CPT I/Y = 11% Result = YTM
The YTM is more than the coupon since the price is less than par.
YTM with Semiannual Coupons
Suppose a bond with a 10% coupon rate and
semiannual coupons, has a face value of
$1000, 20 years to maturity and is selling for
– Is the YTM more or less than 10%?
– What is the semiannual coupon payment?
– How many periods are there?
YTM with Semiannual Coupons
Suppose a bond with a 10% coupon rate and semiannual coupons, has a face value of $1,000, 20 years to maturity and is selling for $1,197.93.
1197.93 PV (negative)
CPT PV 4% (= ½ YTM)
YTM = 4%*2 = 8%
NOTE: Solving a semi-
annual payer for YTM
results in a 6-month yield.
The calculator & Excel
solve what you enter.
The 4% value is the 6-month interest rate. YTM is an annual rate.
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