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Corporate Finance

Fifth Edition

Chapter 6

Valuing Bonds

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Chapter Outline

6.1 Bond Cash Flows, Prices, and Yields

6.2 Dynamic Behavior of Bond Prices

6.3 The Yield Curve and Bond Arbitrage

6.4 Corporate Bonds

6.5 Sovereign Bonds

Learning Objectives (1 of 4)

Identify the cash flows for both coupon bonds and zero-coupon bonds, and calculate the value for each type of bond.

Calculate the yield to maturity for both coupon and zero-coupon bonds, and interpret its meaning for each.

Learning Objectives (2 of 4)

Given coupon rate and yield to maturity, determine whether a coupon bond will sell at a premium or a discount; describe the time path the bond’s price will follow as it approaches maturity, assuming prevailing interest rates remain the same over the life of the bond.

Learning Objectives (3 of 4)

Illustrate the change in bond price that will occur as a result of changes in interest rates; differentiate between the effect of such a change on long-term versus short-term bonds.

Discuss the effect of coupon rate to the sensitivity of a bond price to changes in interest rates.

Define duration, and discuss its use by finance practitioners.

Learning Objectives (4 of 4)

Calculate the price of a coupon bond using the Law of One Price and a series of zero-coupon bonds.

Discuss the relation between a corporate bond’s expected return and the yield to maturity; define default risk and explain how these rates incorporate default risk.

Assess the creditworthiness of a corporate bond using its bond rating; define default risk.

6.1 Bond Cash Flows, Prices, and Yields (1 of 2)

Bond Terminology

Bond Certificate

States the terms of the bond

Maturity Date

Final repayment date

Term

The time remaining until the repayment date

Coupon

Promised interest payments

6.1 Bond Cash Flows, Prices, and Yields (2 of 2)

Bond Terminology

Face Value

Notional amount used to compute the interest payments

Coupon Rate

Determines the amount of each coupon payment, expressed as an A P R

Coupon Payment

Zero-Coupon Bonds (1 of 7)

Zero-Coupon Bond

Does not make coupon payments

Always sells at a discount (a price lower than face value), so they are also called pure discount bonds

Treasury Bills are U.S. government zero-coupon bonds with a maturity of up to one year.

Zero-Coupon Bonds (2 of 7)

Suppose that a one-year, risk-free, zero-coupon bond with a $100,000 face value has an initial price of $96,618.36. The cash flows would be

Although the bond pays no “interest,” your compensation is the difference between the initial price and the face value.

Zero-Coupon Bonds (3 of 7)

Yield to Maturity

The discount rate that sets the present value of the promised bond payments equal to the current market price of the bond

Price of a Zero-Coupon bond

Zero-Coupon Bonds (4 of 7)

Yield to Maturity

For the one-year zero coupon bond:

Thus, the Y T M is 3.5%

Zero-Coupon Bonds (5 of 7)

Yield to Maturity

Yield to Maturity of an n-Year Zero-Coupon Bond

Textbook Example 6.1 (1 of 2)

Yields for Different Maturities

Problem

Suppose the following zero-coupon bonds are trading at the prices shown below per $100 face value. Determine the corresponding spot interest rates that determine the zero coupon yield curve

Maturity	1 Year	2 Years	3 Years	4 Years
Price	$96.62	$92.45	$87.63	$83.06

Textbook Example 6.1 (2 of 2)

Solution

Using Equation. 6.3, we have

Alternative Example 6.1 (1 of 2)

Problem

Suppose that the following zero-coupon bonds are selling at the prices shown below per $100 face value. Determine the corresponding yield to maturity for each bond.

Maturity	1 Year	2 Years	3 Years	4 Years
Price	$98.04	$95.18	$91.51	$87.14

Alternative Example 6.1 (2 of 2)

Solution

Zero-Coupon Bonds (6 of 7)

Risk-Free Interest Rates

A default-free zero-coupon bond that matures on date n provides a risk-free return over the same period

Thus, the Law of One Price guarantees that the risk-free interest rate equals the yield to maturity on such a bond

Risk-Free Interest Rate with Maturity n

Zero-Coupon Bonds (7 of 7)

Risk-Free Interest Rates

Spot Interest Rate

Another term for a default-free, zero-coupon yield

Zero-Coupon Yield Curve

A plot of the yield of risk-free zero-coupon bonds as a function of the bond’s maturity date

Coupon Bonds (1 of 2)

Coupon Bonds

Pay face value at maturity

Pay regular coupon interest payments

Treasury Notes

U.S. Treasury coupon security with original maturities of 1–10 years

Treasury Bonds

U.S. Treasury coupon security with original maturities over 10 years

Textbook Example 6.2 (1 of 2)

The Cash Flows of a Coupon Bond

Problem

The U.S. Treasury has just issued a five-year, $1000 bond with a 5% coupon rate and semiannual coupons. What cash flows will you receive if you hold this bond until maturity?

Textbook Example 6.2 (2 of 2)

Solution

The face value of this bond is $1000. Because this bond pays coupons semiannually, from Eq. 6.1, you will receive a

coupon payment every six months of

Here is the timeline, based on a six-month period:

Note that the last payment occurs five years (10 six-month periods) from now and is composed of both a coupon payment of $25 and the face value payment of $1000.

Six month periods, Cash flow in dollars.

Six month period 0, blank.

Six month period 1, $25.

Six month period 2, $25.

Six month period 3, $25.

Six month period 10, $25 + $1000.

Alternative Example 6.2 (1 of 2)

Problem

Suppose that Procter & Gamble has just issued a 10-year, $1000 bond with a 4% coupon rate and semiannual coupon payments. What cash flows will you receive if you hold the bond until maturity?

Alternative Example 6.2 (2 of 2)

Solution:

Here is the timeline, based on a six-month period:

Note that the last payment is composed of both a coupon payment of $20 and the face value payment of $1000.

Coupon Bonds (2 of 2)

Yield to Maturity

The Y T M is the single discount rate that equates the present value of the bond’s remaining cash flows to its current price

Yield to Maturity of a Coupon Bond

Date, Cash flow.

Date 0, negative P.

Date 1, C P N.

Date 2, C P N.

Date 3, C P N.

Date N, C P N + F V.

Textbook Example 6.3 (1 of 3)

Computing the Yield to Maturity of a Coupon Bond

Problem

Consider the five-year, $1000 bond with a 5% coupon rate and semiannual coupons described in Example 6.2. If this bond is currently trading for a price of $957.35, what is the bond’s yield to maturity?

Textbook Example 6.3 (2 of 3)

Solution

Because the bond has 10 remaining coupon payments, we compute its yield y by solving:

We can solve it by trial-and-error or by using the annuity spreadsheet:

Textbook Example 6.3 (3 of 3)

Blank	N P E R	R A T E	P V	P M T	F V	Excel Formula
Given	10	Blank	Negative 957.35	25	1,000	Blank
Solve for Rate	blank	3.00%	Blank	Blank	Blank	= RATE left parenthesis 10, 25, negative 957.35, 1000 right parenthesis

Therefore, y = 3%. Because the bond pays coupons semiannually, this yield is for a six-month period. We convert it to an A P R by multiplying by the number of coupon payments per year. Thus the bond has a yield to maturity equal to a 6% A P R with semiannual compounding.

Financial Calculator Solution (1 of 9)

Since the bond pays interest semi-annually, the calculator should be set to 2 periods per year.

In order, they are as follows: Gold, P forward slash Y R, 2. A series of calculator keys and corresponding values are displayed. In order, they are as follows: N, 10; I forward slash Y R, 6; P V, negative 957.35; P M T, 25; F V, 1,000.

Alternative Example 6.3 (1 of 2)

Problem

Consider the following semi-annual bond:

$1000 par value

7 years until maturity

9% coupon rate

Price is $1,080.55

What is the bond’s yield to maturity?

Alternative Example 6.3 (2 of 2)

Solution

In order, they are as follows: Gold, P forward slash Y R, 2. A series of calculator keys and corresponding values are displayed. In order, they are as follows: N, 14; I forward slash Y R, 7.5; P V, negative 1,080.55; P M T, 45; F V, 1,000.

Textbook Example 6.4 (1 of 2)

Computing a Bond Price from Its Yield to Maturity

Problem

Consider again the five-year, $1000 bond with a 5% coupon rate and semiannual coupons presented in Example 6.3. Suppose you are told that its yield to maturity has increased to 6.30% (expressed as an A P R with semiannual compounding). What price is the bond trading for now?

Textbook Example 6.4 (2 of 2)

Solution

Given the yield, we can compute the price using Eq.65. First, note that a 6.30% A P R is equivalent to a semiannual rate of 3.15%. Therefore, the bond price is

We can also use the annuity spreadsheet:

Blank	N P E R	R A T E	P V	P M T	F V	Excel Formula
Given	10	3.15%		25	1,000	blank
Solve for P V	blank	Blank	Negative 944.98	blank	blank	= P V left parenthesis 0.0315, 10, 25, 1000 right parenthesis

Financial Calculator Solution (2 of 9)

Since the bond pays interest semi-annually, the calculator should be set to 2 periods per year.

Alternative Example 6.4 (1 of 2)

Problem

Consider the bond in the previous example.

Suppose its yield to maturity has increased to 10%

What is the bond’s new price?

Alternative Example 6.4 (2 of 2)

Solution

6.2 Dynamic Behavior of Bond Prices

Discount

A bond is selling at a discount if the price is less than the face value

Par

A bond is selling at par if the price is equal to the face value

Premium

A bond is selling at a premium if the price is greater than the face value

Discounts and Premiums (1 of 3)

If a coupon bond trades at a discount, an investor will earn a return both from receiving the coupons and from receiving a face value that exceeds the price paid for the bond.

If a bond trades at a discount, its yield to maturity will exceed its coupon rate.

Discounts and Premiums (2 of 3)

If a coupon bond trades at a premium, it will earn a return from receiving the coupons, but this return will be diminished by receiving a face value less than the price paid for the bond.

Most coupon bonds have a coupon rate so that the bonds will initially trade at, or very close to, par.

Discounts and Premiums (3 of 3)

Table 6.1 Bond Prices Immediately After a Coupon Payment

When the bond price is	We say the bond trades	This occurs when
greater than the face value	“above par” or “at a premium”	Coupon Rate > Yield to Maturity
equal to the face value	“at par”	Coupon Rate = Yield to Maturity
less than the face value	“below par” or “at a discount”	Coupon Rate < Yield to Maturity

Textbook Example 6.5 (1 of 2)

Determining the Discount or Premium of a Coupon Bond

Problem

Consider three 30-year bonds with annual coupon payments. One bond has a 10% coupon rate, one has a 5% coupon rate, and one has a 3% coupon rate. If the yield to maturity of each bond is 5%, what is the price of each bond per $100 face value? Which bond trades at a premium, which trades at a discount, and which trades at par?

Textbook Example 6.5 (2 of 2)

Solution

We can compute the price of each bond using Eq.6.5. Therefore, the bond prices are

(trades at a premium)

(trades at par)

(trades at a discount)

Financial Calculator Solution (3 of 9)

In order, they are as follows: Gold, P forward slash Y R, 1. A series of calculator keys and corresponding values are displayed. In order, they are as follows: N, 30; I forward slash Y R, 5; P V, negative 176.86; P M T, 10; F V, 100.

Financial Calculator Solution (4 of 9)

Financial Calculator Solution (5 of 9)

Alternative Example 6.5 (1 of 3)

Problem

Suppose that Procter & Gamble issued a bond that has seven years remaining until maturity, a $1000 face value, and a 4% coupon rate with annual coupon payments. If the current market interest rate is 3%, what is bond’s premium or discount? What if the current market rate is 6%?

Alternative Example 6.5 (2 of 3)

Solution:

At a market rate of 3%, the price of the bond will be

So the premium is

Alternative Example 6.5 (3 of 3)

Solution:

At a market rate of 6%, the price of the bond will be

So the discount is

Time and Bond Prices

Holding all other things constant, a bond’s yield to maturity will not change over time.

Holding all other things constant, the price of discount or premium bond will move toward par value over time.

If a bond’s yield to maturity has not changed, then the I R R of an investment in the bond equals its yield to maturity even if you sell the bond early.

Textbook Example 6.6 (1 of 4)

The Effect of Time on the Price of a Coupon Bond

Problem

Consider a 30-year bond with a 10% coupon rate (annual payments) and a $100 face value. What is the initial price of this bond if it has a 5% yield to maturity? If the yield to maturity is unchanged, what will the price be immediately before and after the first coupon is paid?

Textbook Example 6.6 (2 of 4)

Solution

We computed the price of this bond with 30 years to maturity in Example 6.5:

Now consider the cash flows of this bond in one year, immediately before the first coupon is paid. The bond now has 29 years until it matures, and the timeline is as follows:

Textbook Example 6.6 (3 of 4)

Again, we compute the price by discounting the cash flows by the yield to maturity. Note that there is a cash flow of $10 at date zero, the coupon that is about to be paid. In this case, we can treat the first coupon separately and value the remaining cash flows as in Eq. 6.5:

Note that the bond price is higher than it was initially. It will make the same total number of coupon payments, but an investor does not need to wait as long to receive the first one. We could also compute the price by noting that because the yield to maturity remains at 5% for the bond, investors in the bond should earn a

return of 5% over the year:

Textbook Example 6.6 (4 of 4)

What happens to the price of the bond just after the first coupon is paid? The timeline is the same as that given earlier, except the new owner of the bond will not receive the coupon at date zero. Thus, just after the coupon is paid, the price of the bond (given the same yield to maturity) will be

The price of the bond will drop by the amount of the coupon ($10) immediately after the coupon is paid, reflecting the fact that the owner will no longer receive the coupon. In this case, the price is lower than the initial price of the bond. Because there are fewer coupon payments remaining, the premium investors will pay for the bond declines. Still, an investor who buys the bond initially, receives the first coupon, and then sells it earns a 5% return if the bond’s yield does not change:

Financial Calculator Solution (6 of 9)

Initial Price

Financial Calculator Solution (7 of 9)

Price just after first coupon

Price just before first coupon

Alternative Example 6.6 (1 of 3)

Problem

Suppose that W M T issued a bond that has ten years remaining until maturity, a $1000 face value, and a 3% coupon rate with annual coupon payments. If the current market interest rate is 5%, what is the current price of the bond? What will the price be in 4 years, assuming the current market rate remains unchanged?

Alternative Example 6.6 (2 of 3)

Solution:

At a market rate of 5%, the price of the bond will be:

Alternative Example 6.6 (3 of 3)

Solution:

At a market rate of 5% and 6 years left to maturity, the price of the bond will be:

Figure 6.1 The Effect of Time on Bond Prices

The graph contains four plots, for 10% coupon rate, 5% coupon rate, 3% coupon rate, and zero coupon. The plot for 10% coupon rate is a sawtooth curve that falls from (0, 178) to (30, 100). The plot for 5% coupon rate is a sawtooth curve that extends from (0, 100) to (30, 100). The plot for 3% coupon rate is a sawtooth curve that rises from (0, 70) to (30, 100). The plot for zero coupon is a curve that rises from (0, 22) to (30, 100). All values estimated.

Interest Rate Changes and Bond Prices (1 of 2)

There is an inverse relationship between interest rates and bond prices.

As interest rates and bond yields rise, bond prices fall.

As interest rates and bond yields fall, bond prices rise.

Interest Rate Changes and Bond Prices (2 of 2)

The sensitivity of a bond’s price to changes in interest rates is measured by the bond’s duration.

Bonds with high durations are highly sensitive to interest rate changes.

Bonds with low durations are less sensitive to interest rate changes.

Textbook Example 6.7 (1 of 3)

The Interest Rate Sensitivity of Bonds

Problem

Consider a 15-year zero-coupon bond and a 30-year coupon bond with 10% annual coupons. By what percentage will the price of each bond change if its yield to maturity increases from 5% to 6%?

Textbook Example 6.7 (2 of 3)

Solution

First, we compute the price of each bond for each yield to maturity:

Yield to Maturity	15-Year, Zero-Coupon Bond	30-Year, 10% Annual Coupon Bond
5%	start fraction 100 over 1.05 to the power of 15 end fraction = $48.10	10 times start fraction 1 over 0.05 end fraction left parenthesis 1 minus start fraction 1 over 1.05 to the power of 30 end fraction right parenthesis + start fraction 100 over 1.05 to the power of 30 end fraction = $176.86
6%	start fraction 100 over 1.06 to the power of 15 end fraction = $41.73	10 times start fraction 1 over 0.06 end fraction left parenthesis 1 minus start fraction 1 over 1.06 to the power of 30 end fraction right parenthesis + start fraction 100 over 1.06 to the power of 30 end fraction = $155.06

Textbook Example 6.7 (3 of 3)

The price of the 15-year zero-coupon bond changes by

if its yield to maturity increases from

5% to 6%. For the 30-year bond with 10% annual coupons,

the price change is

Even though the 30-year bond has a longer maturity, because of its high coupon rate, its sensitivity to a change in yield is actually less than that of the 15-year zero coupon bond.

Alternative Example 6.7 (1 of 3)

Problem

The University of Pennsylvania sold $300 million of 100-year bonds with a yield to maturity of 4.67%. Assuming the bonds were sold at par and pay an annual coupon, by what percentage will the price of the bond change if its yield to maturity decreases by 1%? Increases by 2%?

Alternative Example 6.7 (2 of 3)

Solution

Yield decreases by 1%

Price increases by 26.5%

Alternative Example 6.7 (3 of 3)

Solution

Yield increases by 2%

Price decreases by 29.9%

Figure 6.2 Yield to Maturity and Bond Price Fluctuations over Time

The plot has numerous fluctuations in Y T M over 30 years, but tends to stay between extremes of Y T M = 4.0% and Y T M = 6.0%. The graph in panel b shows bond price in percentage of face value, versus year, and has four plots. The plot for price with 4% yield is a broken orange line that rises in a curve from (0, 31) through (15, 59) to (30, 100) The plot for price with 5% yield is a green line that rises in a curve from (0, 24) through (15, 48) to (30, 100). The plot for price with 6% yield is a broken purple line that rises in a curve from (0, 31) through (15, 55) to (30, 100) The plot for actual bond price is a blue line with numerous fluctuations, but it remains almost entirely between the plots for 4% yield and for 6% yield.

6.3 The Yield Curve and Bond Arbitrage

Using the Law of One Price and the yields of default-free zero-coupon bonds, one can determine the price and yield of any other default-free bond.

The yield curve provides sufficient information to evaluate all such bonds.

Replicating a Coupon Bond (1 of 3)

Replicating a three-year $1000 bond that pays 10% annual coupon using three zero-coupon bonds:

The timeline is from 0 to 3 increments of 1.

Coupon bond. First year. $100, Second year. $100. Year 3. $1100

First year zero. First year. $100. Second year zero. Year 2. $100. Third year zero. Year 3. $1100. Zero coupon bond portfolio. Year 1 $100, year 2, $100, and Year 3. $1100

Replicating a Coupon Bond (2 of 3)

Table 6.2 Yields and Prices (per $100 Face Value) for Zero-Coupon Bonds

Maturity	1 year	2years	3 years	4 years
Y T M	3.50%	4.00%	4.50%	4.75%
Price	$96.62	$92.45	$87.63	$83.06%

Replicating a Coupon Bond (3 of 3)

By the Law of One Price, the three-year coupon bond must trade for a price of $1153.

• Zero Coupon Bond, 1 year.

Face Value Required, 100.

Cost, 96.62.

• Zero Coupon Bond, 2 years.

Face Value Required, 100.

Cost, 92.45.

• Zero Coupon Bond, 3 years.

Face Value Required, 1100.

Cost, 11 multiplied by 87.63 = 963.93.

• Total cost, $1153.00.

Valuing a Coupon Bond Using Zero-Coupon Yields

The price of a coupon bond must equal the present value of its coupon payments and face value.

Price of a Coupon Bond

Coupon Bond Yields

Given the yields for zero-coupon bonds, we can price a coupon bond

Blank	N P E R	R A T E	P V	P M T	F V	Excel Formula
Given	3	blank	−1,153	100	1,000	blank
Solve for Rate	blank	4.44%	blank	blank	blank	= RATE(3, 100, −1153, 1000)

Financial Calculator Solution (8 of 9)

In order, they are as follows: Gold, P forward slash Y R, 1. A series of calculator keys and corresponding values are displayed. In order, they are as follows: N, 3; I forward slash Y R, 4.44; P V, negative 1,153; P M T, 100; F V, 1,000.

Textbook Example 6.8 (1 of 3)

Yields on Bonds with the Same Maturity

Problem

Given the following zero-coupon yields, compare the yield to maturity for a three-year, zero-coupon bond; a three-year coupon bond with 4% annual coupons; and a three-year coupon bond with 10% annual coupons. All of these bonds are default free.

Maturity	1 year	2 years	3 years	4 years
Zero- coupon Y T M	3.50%	4.00%	4.50%	4.75%

Textbook Example 6.8 (2 of 3)

Solution

From the information provided, the yield to maturity of the three-year, zero-coupon bond is 4.50%. Also, because the yields match those in Table 6.2, we already calculated the yield to maturity for the 10% coupon bond as 4.44%. To compute the yield for the 4% coupon bond, we first need to calculate its price. Using Eq. 6.6, we have

The price of the bond with a 4% coupon is $986.98. From Eq. 6.5, its yield to maturity solves the following equation:

Textbook Example 6.8 (3 of 3)

We can calculate the yield to maturity using the annuity spreadsheet:

	N P E R	R A T E	P V	P M T	F V	Excel Formula
Given	3		negative 986.98	100	1,000
Solve for Rate		4.47%	Blank			= RATE(3, 40, −986.98, 1000)

To summarize, for the three-year bonds considered

Coupon rate	0%	4%	10%
Y T M	4.50%	4.47%	4.44%

Financial Calculator Solution (9 of 9)

Treasury Yield Curves

Treasury Coupon-Paying Yield Curve

Often referred to as “the yield curve”

On-the-Run Bonds

Most recently issued bonds

The yield curve is often a plot of the yields on these bonds.

6.4 Corporate Bonds

Corporate Bonds

Issued by corporations

Credit Risk

Risk of default

Corporate Bond Yields (1 of 9)

Investors pay less for bonds with credit risk than they would for an otherwise identical default-free bond.

The yield of bonds with credit risk will be higher than that of otherwise identical default-free bonds.

Corporate Bond Yields (2 of 9)

No Default

Consider a one-year, zero-coupon Treasury Bill with a Y T M of 4%.

What is the price?

Corporate Bond Yields (3 of 9)

Certain Default

Suppose now bond issuer will pay 90% of the obligation.

What is the price?

Corporate Bond Yields (4 of 9)

Certain Default

When computing the yield to maturity for a bond with certain default, the promised rather than the actual cash flows are used.

Corporate Bond Yields (5 of 9)

Certain Default

The yield to maturity of a certain default bond is not equal to the expected return of investing in the bond.

The yield to maturity will always be higher than the expected return of investing in the bond.

Corporate Bond Yields (6 of 9)

Risk of Default

Consider a one-year, $1000, zero-coupon bond issued.

Assume that the bond payoffs are uncertain.

There is a 50% chance that the bond will repay its face value in full and a 50% chance that the bond will default and you will receive $900.

Thus, you would expect to receive $950.

Because of the uncertainty, the discount rate is 5.1%.

Corporate Bond Yields (7 of 9)

Risk of Default

The price of the bond will be

The yield to maturity will be

Corporate Bond Yields (8 of 9)

Risk of Default

A bond’s expected return will be less than the yield to maturity if there is a risk of default.

A higher yield to maturity does not necessarily imply that a bond’s expected return is higher.

Corporate Bond Yields (9 of 9)

Table 6.3 Price, Expected Return, and Yield to Maturity of a One-Year, Zero-Coupon Avant Bond with Different Likelihoods of Default

Avant Bond (1-year, zero-coupon)	Bond Price	Yield to Maturity	Expected Return
Default Free	$961.54	4.00%	4%
50% Chance of Default	$903.90	10.63%	5.1%
Certain Default	$865.38	15.56%	4%

Bond Ratings

Investment Grade Bonds

Speculative Bonds

Also known as Junk Bonds or High-Yield Bonds

Table 6.4 Bond Ratings (1 of 2)

Investment Grade Debt

• A lower case a a slash A A A: Judged to be of the best quality. They carry the smallest degree of investment risk and are generally referred to as “gilt edged.” Interest payments are protected by a large or an exceptionally stable margin and principal is secure. While the various protective elements are likely to change, such changes as can be visualized are most unlikely to impair the fundamentally strong position of such issues.

• A lowercase a slash A A: Judged to be of high quality by all standards. Together with the A a a group, they constitute what are generally known as high-grade bonds. They are rated lower than the best bonds because margins of protection may not be as large as in A a a securities or fluctuation of protective elements may be of greater amplitude or there may be other elements present that make the long-term risk appear somewhat larger than the A a a securities.

• A slash A: Possess many favorable investment attributes and are considered as upper-medium-grade obligations. Factors giving security to principal and interest are considered adequate, but elements may be present that suggest a susceptibility to impairment sometime in the future.

• B lowercase a a slash B B B: Are considered as medium-grade obligations (i.e., they are neither highly protected nor poorly secured). Interest payments and principal security appear adequate for the present but certain protective elements may be lacking or may be characteristically unreliable over any great length of time. Such bonds lack outstanding investment characteristics and, in fact, have speculative characteristics as well.

Table 6.4 Bond Ratings (2 of 2)

[Table 6.4 continued]

*Ratings: Moody’s/Standard & Poor’s

Source:

www.moodys.com

Speculative Bonds

• B a slash B B Judged to have speculative elements; their future cannot be considered as well assured. Often the protection of interest and principal payments may be very moderate, and thereby not well safeguarded during both good and bad times over the future. Uncertainty of position characterizes bonds in this class.

• B slash B: Generally, lack characteristics of the desirable investment. Assurance of interest and principal payments of maintenance of other terms of the contract over any long period of time may be small.

• C lowercase a a slash C C C: Are of poor standing. Such issues may be in default or there may be present elements of danger with respect to principal or interest.

• C a slash C C: Are speculative in a high degree. Such issues are often in default or have other marked shortcomings.

• C slash C, D: Lowest-rated class of bonds, and issues so rated can be regarded as having extremely poor prospects of ever attaining any real investment standing.

Corporate Yield Curves

Default Spread

Also known as Credit Spread

The difference between the yield on corporate bonds and Treasury yields

Figure 6.3 Corporate Yield Curves for Various Ratings, February 2018

Source: Bloomberg

The U S Treasury’s yield curve is blue, and it rises at a decreasing rate from (0.25, 2.3%) through (10, 2.8%) to (30, 3,2%). The U S Industrials, Ay Ay Ay curve is yellow, and it rises at a decreasing rate from (0.25, 2.5%) through (10, 3.5%) to (30, 3.9%). The U S Industrials, Ay curve is green, and it rises at a decreasing rate from (0.25, 2.7%) through (10, 3.7) and (20, 4.2) to (30, 4.5%). All values estimated.

Figure 6.4 Yield Spreads and the Financial Crisis

Source:

Bloomberg.com

The graph has three plots with similar fluctuations. The plot for Ay ay ay is blue, and begins at (2005, 0.6), peaks at (2009, 2.5) and (2012, 1) and ends at (2018, 0.7). The plot for Ay is red, and begins at (2005, 1), peaks at (2009, 4) and (2012, 1.5) and ends at (2018, 1.0). The plot for B ay ay is green, and begins at (2005, 1.5), peaks at (2009, 6.7) and (2012, 2.4) and ends at (2018, 1.3). The graph in panel b shows yield spread of short-term loans to major international banks, L I B O R, versus U S treasury bonds, with spread in percentage versus time, measured on January fourth of each year from 2005 to 2018. The plot remains below 1% until mid-2007, then begins fluctuating between 1% and 2.5% until mid-2007. The plot then spikes steeply to 4.6% in late 2008, but drops below 1% in mid-2009, remaining below 1% through 2018. All values estimated.

6.5 Sovereign Bonds

Bonds issued by national governments

U.S. Treasury securities are generally considered to be default free.

All sovereign bonds are not default-free,

e.g., Greece defaulted on its outstanding debt in 2012.

Importance of inflation expectations.

Potential to “inflate away” the debt.

European sovereign debt, the E M U, and the E C B

Figure 6.5 Percent of Debtor Countries in Default or Restructuring Debt, 1800–2006

Source: Data from This Time Is Different, Carmen Reinhart and Kenneth Rogoff, Princeton University Press, 2009.

The data depicted is as follows:

• 1800 to 1830: 1 to 15 percent

• 1840 to 1880: 15 to 50 percent

• 1890 to 1930: 10 to 20 percent

• 1940 to 1950: 20 to 45 percent

• 1960 to 1980: 5 to 10 percent

• 1990 to 2000: 10 to 35 percent

Figure 6.6 European Government Bond Yields, 1976–2018

Source: Federal Reserve Economic Data,

research.stlouisfed.org/fred2 .

The graph has six plots representing Germany, Ireland, Spain, France, Portugal, and Italy. The plot for Germany begins at (1976, 8.2); peaks at (1982, 19), (2012, 11.8); (1995, 7.5) and ends at (2016, 0.5). The plot for France begins at (1976, 10.1); peaks at (1982, 17), (1991, 10.5); (1995, 8.1) and ends at (2016, 1). The plot for Ireland begins at (1976, 14); peaks at (1982, 19), (1987, 13.5); (1993, 10.5); (2012, 12) and ends at (2016, 1). The plot for Spain begins at (1980, 15.5); peaks at (1984, 18), (1987, 13.5); (1991, 15); (1995, 12); (2012, 7) and ends at (2016, 2). The plot for Italy begins at (1991, 13.8); peaks at (1992, 14.4), (1995, 13.5); (2012, 7) and ends at (2016, 1.8). All values estimated.

Chapter Quiz

What is the relationship between a bond’s price and its yield to maturity?

If a bond’s yield to maturity does not change, how does its cash price change between coupon payments?

How does a bond’s coupon rate affect its duration—the bond price’s sensitivity to interest rate changes?

Explain why two coupon bonds with the same maturity may each have a different yield to maturity.

There are two reasons the yield of a defaultable bond exceeds the yield of an otherwise identical default-free bond. What are they?

Corporate Finance

Fifth Edition

Chapter 6

Appendix

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101

Forward Interest Rates

6A.1 Computing Forward Rates

A forward interest rate (or forward rate) is an interest rate that we can guarantee today for a loan or investment that will occur in the future.

In this chapter, we consider interest rate forward contracts for one-year investments, so the forward rate for year 5 means the rate available today on a one-year investment that begins four years from today.

Computing Forward Rates (1 of 5)

By the Law of One price, the forward rate for year 1 is equivalent to an investment in a one-year, zero-coupon bond.

Computing Forward Rates (2 of 5)

Consider a two-year forward rate.

Suppose the one-year, zero-coupon yield is 5.5% and the two-year, zero-coupon yield is 7.0%.

We can invest in the two-year, zero-coupon bond at 7.0%

and earn

after two years.

Or, we can invest in the one-year bond and earn $1.055 at the end of the year.

We can simultaneously enter into a one-year interest

rate forward contract for year 2 at a rate of

Computing Forward Rates (3 of 5)

At then end of two years, we will have

Since both strategies are risk free, by the Law of One Price, they should have the same return:

Computing Forward Rates (4 of 5)

Rearranging, we have

The forward rate for year 2 is

Computing Forward Rates (5 of 5)

In general,

Rearranging, we get the general formula for the forward interest rate:

Textbook Example 6A.1 (1 of 2)

Computing Forward Rates

Problem

Calculate the forward rates for years 1 through 5 from the following zero-coupon yields:

Maturity	1	2	3	4
Y T M	5.00%	6.00%	6.00%	5.75%

Textbook Example 6A.1 (2 of 2)

Solution

Using Eqs. 6A.1 and 6A.2:

Alternative Example 6.A1 (1 of 2)

Problem

At the end of 2015, yields on Canadian government bonds were

Maturity	1 Year	2 Years	3 Years	4 Years	5 Years	6 Years	7 Years	8 Years	9 Years	10 Years
Yield	0.51%	0.47%	0.54%	0.67%	0.82%	0.97%	1.13%	1.28%	1.42%	1.55%

Based on this yield curve, what is

Alternative Example 6.A1 (2 of 2)

Solution

6A.2 Computing Bond Yields from Forward Rates

It is also possible to compute the zero-coupon yields from the forward interest rates:

For example, using the forward rates from Example 8A.1, the four-year zero-coupon, yield is

6A.3 Forward Rates and Future Interest Rates (1 of 2)

How does the forward rate compare to the interest rate that will actually prevail in the future?

It is a good predictor only when investors do not care about risk.

Textbook Example 6A.2 (1 of 2)

Forward Rates and Future Spot Rates

Problem

JoAnne Wilford is corporate treasurer for Wafer Thin Semiconductor. She must invest some of the cash on hand for two years in risk-free bonds. The current one-year, zero-coupon yield is 5%. The one-year forward rate is 6%. She is trying to decide between three possible strategies: (1) buy a two-year bond, (2) buy a one-year bond and enter into an interest rate forward contract to guarantee the rate in the second year, or (3) buy a one-year bond and forgo the forward contract, reinvesting at whatever rate prevails next year. Under what scenarios would she be better off following the risky strategy?

Textbook Example 6A.2 (2 of 2)

Solution

From Eq. 6A.3, both strategies (1) and (2) lead to the same

risk-free return of

The third strategy returns

where r is the one-

year interest rate next year. If the future interest rate turns out to be 6%, then the two strategies will offer the same return. Otherwise Wafer Thin Semiconductor is better off with strategy (3) if the interest rate next year is greater than the forward rate—6%—and worse off if the interest rate is lower than 6%.

Forward Rates and Future Interest Rates (2 of 2)

We can think of the forward rate as a break-even rate.

Since investors do care about risk,

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117

Coupon RateFace Value

Number of Coupon Payments per Year

CPN

(1)

YTM

100,000

96,618.36

(1 + )

YTM

+==

100,000

11.035

96,618.36

YTM

æö

ç÷

èø

YTM

==-=

æö

==-=

ç÷

èø

æö

==-=

ç÷

èø

æö

==-=

ç÷

èø

100

13.50%

96.62

100

14.00%

92.45

100

14.50%

87.63

100

14.75%

83.06

rYTM

=-==

æö

=-==

ç÷

èø

æö

=-==

ç÷

èø

æö

=-==

ç÷

èø

100

10.022%

98.04

100

10.0252.5%

95.18

100

10.033%

91.51

100

10.0353.5%

87.14

YTM

rYTM

=´=

CPN$1000$25.

æö

=´-+

ç÷

èø

(1)(1)

PCPN

yyy

æö

=´-+

ç÷

èø

1010

111000

957.3525 1

(1)(1)

yyy

957.35

= RATE 10,25,

-957.35,

(

1000

)

=´=

N 7 years214

=´¸=

PMT(9%$1000)2$45

æö

=´-+=

ç÷

èø

1010

111000

251$944.98

0.03151.03151.0315

944.98

= PV 0.0315,10,

25,

(

1000

)

=´=

N7 years214

=´¸=

PMT(9%$1000)2$45

æö

=´-+=

ç÷

èø

3030

11100

(10% coupon)101$176.86

0.051.051.05

æö

=´-+=

ç÷

èø

3030

11100

(5% coupon)51$100.00

0.051.051.05

æö

=´-+=

ç÷

èø

3030

11100

(3% )31$69.26

0.051.051.05

Pcoupon

$1062.30$1000$62.30.

$1000$888.35$111.65.

æö

=´-+=

ç÷

èø

3030

11100

101$176.86

0.051.051.05

æö

=+´-+=

ç÷

èø

2929

11100

(just before first coupon)10101$185.71

0.051.051.05

´=

$176.861.05$185.71.

æö

=´-+=

ç÷

èø

2929

11100

(just after first coupon)101$175.71

0.051.051.05

(10175.71)

1.05.

176.86

$175.71$10$185.71

100

$48.10

1.05

æö

´-+=

ç÷

èø

3030

11100

101$176.86

0.051.051.05

100

$41.73

1.06

æö

´-+=

ç÷

èø

3030

11100

101$155.06

0.061.061.06

(41.7348.10)

13.2%

48.10

(155.06176.86)

12.3%.

176.86

$1,265.06

10.265

$1,000

-=-

$700.62

10.299

$1,000

(

)

Bond Cash Flows

VPV

=++¼+

+++

1(1)(1)

CPNCPNCPNFV

YTMYTMYTM

=++=

1001001001000

$1153

1.0351.041.045

==++

+++

1001001001000

1153

(1)(1)(1)

yyy

=++=

1001001001000

$1153

1.04441.04441.0444

=++=

4040401000

$986.98

1.0351.041.045

=++

+++

4040401000

$986.98

(1)(1)(1)

yyy

986.98

===

10001000

$961.54

1 1.04

YTM

===

900900

$865.38

1 1.04

YTM

=-=-=

1000

1115.56%

865.38

YTM

900

1.04

865.38

950

$903.90

1.051

=-=-=

1000

1110.63%

903.90

YTM

fYTM

$(1.07)

(

)

(

$1.0551

)

(1.07)(1.055)(1)

+==

1.07

(1)1.0852

1.055

8.52%.

+=++

-1

(1)(1)(1)

nnn

YTMYTMf

(1)

YTM

=-=-=

5.00%

(1)

1.06

117.01%

(1)1.05

(1)

1.06

116.00%

(1)106

(1)

1.0575

115.00%

(1)1.06

fYTM

YTM

210

=-==

(1.0047)

10.00430.43%

(1.0051)

(1.0155)

10.027282.27%

(1.0142)

+´+´´+=+

fffYTM

(1)(1)...(1)(1)

[

]

[

]

+=++++

41234

1(1)(1)(1)(1)

(1.05)(1.0701)(1.06)(1.05)

1.0575

YTMffff

+=++=

212

(1)(1)(1)(1.05)(1.06).

YTMYTMf

(

)

1.051

(

)

Expected Future Spot Interest Rate = For

ward Interest Rate + Risk Premium

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