Relationship Between Coupon and Yield

Bond Prices: Relationship Between Coupon and Yield

• If YTM = coupon rate, then par value = bond price

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• 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.

 

 

 

 

Slide 11

 

6-11

The Bond-Pricing Equation Adjusted for Semi-annual Coupons

2t

2t

YTM/2)(1

F

YTM/2

YTM/2)(1

1 -1

2

C ValueBond

+ +

   

   

+ =

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

 

 

 

 

 

Slide 12

 

6-12

Example

• 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?

 

 

 

 

 

 

Slide 13

 

6-13

Semiannual Bonds

• 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

 

 

 

 

 

Slide 14

 

6-14

Example 7.1

• 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

2t

2 YTM1

F

2 YTM

2 YTM1

1 -1

2 C Value Bond

+ +

    

    

+ =

14

14

)08.1(

1000

08.0

)08.1(

1 1

70B +

   

   

 −

=

Using the calculator:

14 N

8 I/Y

70 PMT

1000 FV

CPT PV = -917.56

 

 

 

 

 

 

Slide 15

 

6-15

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

long time.

 

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.

 

 

Slide 16

 

6-16

Figure 7.2

 

 

 

 

 

 

Slide 17

 

6-17

• 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

 

 

 

 

 

 

Slide 18

 

6-18

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%?

15 N

928.09 PV (enter as a negative)

1000 FV

100 PMT

CPT I/Y = 11% Result = YTM

 

 

The YTM is more than the coupon since the price is less than par.

 

 

 

 

 

Slide 19

 

6-19

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

$1197.93.

– Is the YTM more or less than 10%?

– What is the semiannual coupon payment?

– How many periods are there?

 

 

 

 

 

 

Slide 20

 

6-20

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.

40 N

1197.93 PV (negative)

1000 FV

50 PMT

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.

 

 

 

 

 

Slide 21

 

6-21


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