2 Introduction to Bonds

One of the most important uses of time value of money is asset valuation. Asset valuation, in layman’s terms, simply means determining how much you should pay for an asset if you want to buy the asset, and how much you should charge if you want to sell it.

The first asset we will consider is bonds. As with any new topic, we start with the topic’s unique vocabulary. Next you will walk through the process of how and why a bond comes into existence. Investors buy assets for the financial return the assets provide them. We will see the different ways bonds provide their owners return, and the risks that bond owners experience from owning a bond over the bond’s life.

What is a bond?

A bond is a financial instrument (a financial product) used to borrow money from investors. In the bond relationship, the sellers of the bonds (who we call the bonds “issuers”) are the borrowers, and the buyers of the bonds are the lenders (who we call investors). Issuing a bond is an alternative to borrowing from a bank.

A bond is known as a fixed income security because the promised cash-flows are stated in the terms (description) of the bond.

Who uses bonds to borrow?

Governments and businesses use bonds to borrow from investors. The US Federal Government issues (sales) bonds to investors to pay for the Federal Government’s budget deficit. The budget deficit is just the difference between how much Congress has decided to spend and how much tax revenues it has collected. Since these bonds are issued by the US Treasury, they are known as Treasury Securities. The treasury securities have different names depending on their maturity and the way they provide return to investors. The Treasury issues Treasury Bills, Treasury notes and Treasury Bonds, or T-Bills, T-Notes and T-Bonds, as they are often called.

State and local governments also borrow money from investors by issuing bonds. These bonds are called “Muni” bond, short for municipal bonds. These bonds are used by local governments to pay for infrastructure such as roads, sewers, school buildings and facilities, etc.

Businesses also use bonds to borrow from investors. These bonds are referred to as “corporate bonds.” Firms issue corporate bonds as part of their overall capital structure to have money available to pay for and maintain their plant and equipment. The interest rate and features of these bonds depends on the particular firm who issued the bond.

Bond Vocabulary

We know that governments and firms issue bonds to investors. When the investor buys the bond, this establishes a contractual relationship between the bond issuer and the investor. The investor pays an amount to the bond issuer which is the bond’s price. In exchange, the bond issuer agrees to pay periodic cash-flows (usually semiannually) called coupons to the investor. In addition, the bond issuer agrees to pay an additional lump sum cash-flow, called the face value, at some pre-specified future date, called the maturity date. The face value is also known as par value. The maturity date is the last date the firm pays any cash-flows to the investor. In essence, the bond ceases to exist after the maturity date. The size of the coupon can be stated in dollar terms or as a percentage of the face value. When it is stated as a percentage of face value the percentage is known as the coupon rate. These concepts and terms are illustrated in the figure below.

This figure illustrates the cash-flows for the following bond:

-   2-year maturity
-   Semiannual coupon (every six months)
-   Face Value of $1,000
-   Coupon of $50 (coupon rate of 5% = $50/$1000)
-   Price Today of $1,000
  1. Today the investor pays the firm $1,000,
  2. Six months later the firm pays the investor the first coupon of $50.
  3. and 4. The firm pays the second and third coupons at one year and 1 ½ years.
  4. The firm pays the last coupon and the face value.

Now that you know basic bond terminology, watch the video “What is a bond” for additional clarification.



Click here for: In-Class Exercise 11: Bond Terms and Concepts

2.1 Bond Valuation

What’s the price of a bond?

Based on modern finance principles the price of any asset is the present value of the cash-flows the owner of the asset is expected to get from owning the asset at a particular time. Since a bond is an asset to its owners, this principle also applies to bonds.

We have two inputs to calculate the price of any asset

The expected cash-flows  

The discount rate.

Since a bond is a fixed income security, its expected cash-flows are stated in the bond terms. The discount rate on an asset depends on the riskiness of the asset. The riskier the asset is the higher the discount rate used on the asset’s cash-flows. For now, we will just take the discount rate as given.

Let’s demonstrate calculating the price of a typical bond with the following features.

  • N-year maturity

  • Annual coupons

  • Face Value of $FV

  • Coupon of size $C

  • Price today of \(P_{bond_{0}}\)

Returning back to your knowledge of time value of money, the price of a bond would be

\[P_{bond_0} = \frac{\$C}{(1+R)^1}+\frac{\$C}{(1+R)^2}+...+\frac{\$C}{(1+R)^N}+\frac{\$FV}{(1+R)^N}\]

where $C is the dollar coupon, R is the discount rate, $FV is the face value and N is the number of years until maturity.

The price is just the present value of the expected cash-flows the owner of the bond will receive discounted at a rate R.

Let’s do a numerical example. What is the price of a bond with the following features?

• N-year maturity = 4 years • Annual coupons = coupon paid once per year • Face Value of FV = $1,000 • Coupon of size C = $60 • Discount rate = 10%

Apply the present value formula

The price of this bond would be $873.21 when the discount rate (interest rate) is 10%.

What is the relationship between a bond’s price and the interest rate?

Now that you know how to calculate the price of a bond, what would happen to the price of the above bond if the interest rate decreased to 6%? You know how to figure this out. Just use the equation for the price of a bond with 6% instead of 10%. If you do that, you will see

How about the bond price when the interest rate equals 4%? Just repeat the price calculation again, but with the interest rate equal to 4%.

You’ve just illustrated a VERY IMPORTANT relationship between interest rate and bond prices. Interest rates and bond prices move in opposite directions.

Play with this interactive widget to see how the price-yield relationship works. As you change the bond’s maturity, you’ll see the curvature of the line change. This indicates changes in the sensitivity of the bond price to changes in the yield. You can also change the bond’s coupon rate and the yield (discount rate), which will change the bond’s price.

Premium, Par or Discount?

We saw that the price of a bond moves in opposite directions from the interest rate. Let’s summarize the calculations from the bond above in a table and look for some more details. Before we do, what was the coupon rate of the bond? Recall the coupon rate is ($Coupon/$Face Value), so for the bond above the coupon rate was ($60/$1,000) = 0.06 = 6%.

• Maturity = 4 years

• Annual coupons

• Face Value of FV = $1,000

• Coupon of size = $60

When the interest rate was 10%, the bond has a price of $873.21. Note that this price is lower than the par value of the bond (which equals $1,000. Remember face and par are equal to each other). When the price is lower than par, we say the bond is selling “at a discount.”

When the interest rate was 6% we see the bond price is $1,000 or equal to par. In this circumstance, we say the bond is selling “at par.” Finally, when the interest rate is 4%, the bond price is $1,072.60 which is greater than par, so we say the bond is selling “at a premium.”

One general rule to note from this example is that

Coupon Rate > Interest Rate –> Bond sells “at a premium”

Coupon Rate = Interest Rate –> Bond sells “at par”

Coupon Rate < Interest Rate –> Bond sells “at a discount”

The video “Bond Valuation” will walk you through some bond valuation examples.



Click here for: Worked Problems 9: Bond Valuation

Click here if you want to navigate to the end of the chapter, where you can get some practice calculating the price of a bond using the practice questions widget.

2.2 Semiannual Coupons

You learned how to adjust an Annual Percentage Rate (APR) for monthly compounding when you did the time value of money unit. We want to make a similar adjustment for the fact that many bonds pay coupons semiannually (every six months). The process is mechanical and is best illustrated with an example.

With semiannual coupons, we need to make 3 adjustments to do the calculations
1. Multiply the annual payments by 2 (to take into account the 2 semiannual periods per year)
2. Divide the YTM by 2 (to make the discount rate a 6 month rate)
3. Divide the dollar coupon by 2 (to take into account that the coupon rate is for a whole year)

Consider a bond with the following terms

    Bond Features
    Maturity        5 years
    Coupon Rate     3%
    Face Value      $1,000
    YTM             4%
    Semiannual Coupons
    

What is the bond’s price?

N = 10 (5 years x 2), I/Y = 2 (4%/2), PMT = $15 (0.03x$1,000/2), FV = $1,000

With this information we see that the price of the bond is $955.09

Another adjustment you must make when dealing with semiannual coupon payments is when you want to find the YTM (on an annual basis).

Once again, consider the bond with the following features.

    Bond Features
    Maturity        5 years
    Coupon Rate     4%
    Face Value      $1,000
    Bond Price      $995
    Semiannual Coupons

What is this bonds YTM stated as an annual rate?

Using our financial calculators we would input:

N = 10 (5 years x 2), PV = -$995, PMT = $20 (0.04x$1000/2), FV = $1,000

When we solve for I/Y we get: I/Y = 2.0558% , but to make this stated as an annual rate we need to multiply this by 2 to get 4.1117% as the annual YTM.

Now that you’ve seen the basics, watch the video below to see additional examples with semi-annual compounding



Click here if you want to navigate to the end of the chapter, where you can get some practice calculating the price of a bond with semiannual coupon payments using the practice questions widget.

2.3 Bond Return to the Investor

Investors buy assets for the returns the assets provide them. Every asset has the potential to provide its owner with return from two sources

  1. Yield

  2. Capital gains/losses

The yield on an asset generally refers to the periodic payments an asset provides its owner. For example, if you own rental property, as the landlord you would get monthly rent payments from your tenants. Another example would include the monthly interest you get from your bank savings account.

Capital gains refers to the case where the owner of the asset buys the asset at one price, and then later sells that asset (hopefully) at a higher price. The difference in the buy and sell price is the capital gain (or loss if negative).

A bond can provide its owner with both of these types of return. Let’s return to the cash-flow diagram to illustrate.

The market interest rate is 6%. The bond’s prices over its life are shown, as well as the coupons and the face value the bond pays. The bond’s return, between each coupon date over the life of the bond, is shown in the table.

The income from the bond, called the current yield is shown in the first row of the table. The current yields is given by the formula

\[Current\;Yield\;(from\;period\;N\;to\;N+1)=\frac{\$Coupon_{N+1}}{\$Price_N}\times100\%\]

or in words, “the coupon in the next period divided by the price in the previous period.”

For example, the coupon rate from period 1 to period 2 is equal to 5.14% which is the $50 coupon in year 2 divided by the price from year 1 ($973.27). In economic terms this means that the owner of the bond has an asset that is worth $973.27 in year 1 that pays her a coupon of $50 in year 2. This income of $50 represents a percentage return of 5.14%.

In addition, if she owned the bond in period 1 and sold it in period 2, she would make a capital gain of ($981.67 -$973.27) = $8.40 or in percentage terms 0.86%.

Her total return from owning the bond from period 1 to 2 would be 6% (5.41%+0.86%). Notice that this is also the bond’s yield to maturity, YTM and the discount rate. (you should be able to verify that the YTM is 6% using the bond’s terms).

This is an important feature: When the YTM stays constant over the bond’s life, the total return of the bond between each period (coupon date) will equal the bond’s YTM.

Watch the mini-lecture for additional explanations and examples regarding bond returns. Current Yield Capital Gains and Total Returns



Click here for: Worked Problems 10: Bond Return to the Investor

Click here if you want to navigate to the end of the chapter, where you can get some practice calculating the current yield and capital gains yield of a bond using the practice questions widget.

2.4 Yield to Maturity and Yield to Call

A bond’s yield to maturity or YTM is an interest rate that has the following three features.

The YTM is
1. The interest rate that equates the present value of the bond’s cash-flows with the bond’s price
2. The per period return the investor would earn if she held the bond to its maturity AND the interest rate remained at the YTM during the bond’s life
3. An alternative way to indicate the bond’s price.

First the YTM is the interest rate that equates the present value of the bond’s cash-flows with the bond’s price. Consider the bond with the features in the table

    Bond Features
    Maturity        4 years
    Coupon Rate     4%
    Face Value      $1,000
    Dollar Coupons  $40.00
    Bond Price      $965.35
    

The YTM is the interest rate relates to the price of the bond as shown.

where we find that the YTM = 4.98%.

Next the YTM is the per period return the investor would earn if she held the bond to its maturity AND the interest rate remained at the YTM during the bond’s life.

To see this use what you’ve learned to trace how much you would have if you purchased this bond today for $965.35, held it to maturity and reinvested the coupons along the way at the bond’s YTM of 4.98%.

Now let’s calculate your per period rate of return over the 4 years you held the bond. You purchased the bond today for $965.35 and four years later you have $1,172.35 as a result of reinvesting the coupons and getting the face value.

N = 4, PV = -$965.35, PMT = 0, FV = $1,172.35 –> I/Y = ?

I/Y is 4.98% so you did in fact earn the YTM by holding the bond over its life and reinvesting the coupons. Note however this result depends on the interest rate remaining at 4.98% over the life of the bond.

Finally, the YTM serves as an alternative way to inform the investor of the bond’s price. Since the promised cash-flows are fixed, if you know the YTM of a bond, you can calculate its price and visa-versa.

2.4.1 Yield to Call (YTC)

The next topic we cover in this unit is the idea of a “callable” bond. When a firm issues a typical bond, it must make the cash-flows defined in the bond’s terms or it will be considered in default. This is true no matter what has happened to the market interest rates since the bond was first issued.

For example, let’s say you purchased the bond with the following features

    Bond Features
    Maturity        6 years
    Coupon Rate     4%
    Face Value      $1,000
    Dollar Coupons  $40.00
    Bond Price      $1,000

Based on your knowledge of bonds, you know that because the bond price is at par, the market interest rate must be 4% (remember: the bond price equals par when the coupon rate equals the market interest rate). Over the life of the bond, the interest rate may change, and you know that these interest rate changes affect the investor.

Changes in the interest rate also provide opportunities for the company that issued the bond. Say the interest rate has gone down (for example, to 1%) after this bond was issued. Under these circumstances, the company would like to issue a new bond at 1% and buy back the bonds it has issued at 4%. By buying back the old bonds, the firm would lower the coupons it would have to pay investors from $40 to $10.
This is analogous to when a homeowner refinances her mortgage. Assume she first took out a home loan at 4%, with mortgage payments of $1,200 per month. Then, a year later interest rates on home loans have dropped to 1%. She will now “refinance” her mortgage which means she takes out a new loan at 1% to pay off her first loan. She does this because at the new lower interest rate, her monthly payments will be lower.

The same process is possible for a firm that issues a bond IF the bond has a “Call Feature.” With a “callable” bond, the bond has an additional feature that the firm may buy the bond back from the investor at (or after) a certain date (called the call date), for a predetermined price (called the call price) which is face value plus an additional amount called the “call premium.” The owner must sell the bond to the firm at the call price regardless of the bond’s actual market price. When the bond is called away from the investor, the investor must sell the bond back to the firm at a price below what the bond is actually worth. Clearly, this hurts the investor and benefits the firm.

Let’s walk through an example with the following bond.

    Maturity                    6 years
    Coupon Rate                 4%
    Face Value                  $1,000
    Dollar Coupons              $40.00
    Bond Price when issued      $1,000
    Call date                   4 years
    Call premium                $40

We will only examine two possibilities:

  1. the investor holds the bond until maturity

  2. the firm calls the bond away from the investor at year 4.

The cash-flows to the investor under the two cases are:

The yield to call (YTC) is just the ~per period rate of return~ that the investor would earn using the cash-flows when the bond is called. In this case,

N = 4, PV = -$1,000, PMT = $40, FV = $1,040 –> I/Y = 4.92%.

Note, the investor does not get two coupons as a result of the bond being called away from her. This loss of coupon income is bad for the investor.

For simplicity, in the example above, we did not take into account that since our investor was buying a bond with a call feature, she is aware that the bond MAY be called away from her, and she would not get the last two coupons. Because of this chance, she would only be willing to buy the callable bond at a price that is lower than a similar bond that does not have a call feature.

A bond with a call feature will sell at a lower price than a similar bond that does not have a call feature.

To Call or Not to Call?

A firm will call back a bond at the call price if the market interest rate on (or after) the call date has dropped to a level such that the market price of the bond is greater than the call price. Let’s illustrate with an example.

XYZ firm has sold a bond with the following features.

    Maturity                    6 years
    Coupon Rate                 4%
    Face Value                  $1,000
    Dollar Coupons              $40.00
    Bond Price when issued      $985
    Call date                   4 years
    Call premium                $40

Assume that on the call date, the market interest rate is 2%. Should XYZ call back the bond and if it does, how much does it save the company?

At the call date the bond has the remaining cash-flows.

The interest rate is 2.0%, therefore the market price of the bond is $1,038.83 (N = 2, I/Y=2, PMT = $40, FV= $1,000, PV –> $1,038.83). The firm has the right under the call terms of the bond to buy back the bond from the investor for a price of $1040.

The firm would NOT call back the bond. The firm will not pay $1,040 for something that is only worth $1,038.83, otherwise the firm would be losing $1.17.

However, if the interest rate on the call date was 1%, the market price of the bond would be $1,059.11 (N = 2, I/Y=1,PMT = $40, FV= $1,000, PV –> $1,059.11) and the firm WOULD call back the bond. The firm is able to buy from the investor something that is worth $1,059.11 for only $1,040. The firm would gain $19.11 by calling the bond.

Watch the video below to see further examples of YTM and YTC



Click here for: Worked Problems 11: YTM and YTC

Click here for: In-Class Exercise 12: Bond Calculations – Price, Yield to Maturity and Yield to Call

Click here if you want to navigate to the end of the chapter, where you can get some practice calculating the YTM and YTC of a bond using the practice questions widget.

2.5 Bond Risk – Default, Price Risk and Reinvestment Risk

Every asset faces various types of risk. Most bonds face at least three types of risk

  1. Default risk

  2. Price risk

  3. Reinvestment risk

What is default risk?

Default risk refers to the possibility that the issuer of the bond will be unable to make the promised cash payments (either coupons, the face value, or both) on the scheduled payment date. The higher the perceive default risk the lower the amount investors will pay for a bond and the higher the YTM on the bond will be. Many bonds are given a “rating” that indicates a rating agency’s opinion on how likely a bond from a particular issuer is to default. The three dominant rating agencies are Standard and Poor’s, Moody’s, and Fitch Group. The ratings are just a professional opinion from each agency and there is no guarantee that a bond with a good rating will not default.

Bonds issued by the US Treasury (T-bonds, T-Notes and T-Bills) are generally consider to have such a low probability of default that we use the interest rates on these bonds as the default free interest rate associated with each investment horizon (maturity date).

Municipal bonds, (bonds issued by state and local governments) face greater default risk than Treasury securities, and depending on the particular state or city, can have a wide range of credit ratings. Corporate bonds are the third general category of bonds we consider. These are bonds issued by individual companies. Similar to municipal bonds, the default risk associated with a particular corporate bond depends on the financial strength of the company.

2.5.1 Price risk and Reinvestment Risk

Two additional kinds of risk that bonds face are price risk and reinvestment risk. Price risk refers to the fact that bond prices change when market interest rates change. Reinvestment risk refers to the fact that when bond coupons are received they are reinvested at the prevailing market interest rate when they are received. If the market interest changes, the accumulated income from reinvesting the coupons will change depending on whether the interest rate increased or decreased.

To illustrate price and reinvestment risk, let’s trace the price of a bond under two possible interest rates: 3% and 2%.

We have a bond with the following terms.

    Maturity                    4 years
    Coupon Rate                 4%
    Face Value                  $1,000
    Dollar Coupons              $40.00

The cash-flows for such a bond are given as shown.

By now you know how to calculate the price of a bond at any coupon date depending on the interest rate.

When the interest rate is 3%, the price of the bond over the bond’s life is:

If you bought the bond when the interest rate was 3% and then the interest rate changed to 2%, you would experience a “PRICE EFFECT” in each year based on the interest rate change. Specifically, the price effect in each year would be the difference in the price when the interest rate was 3% compared to when the interest rate was 2%.

Now let’s examine the reinvestment effect.

Consider the first row of the table below. The first coupon is received in year 1.

At this point you have $40 worth of accumulated coupon income.

You invest this coupon at the going market rate of 3%. One year later that $40 has grown to $41.20 (= 1.03*$40). In addition, in year 2 you receive the second coupon (see row two) for a total amount of total accumulated income of $81.20 in year 2. The amount each coupon grows to is shown in the table along with the total income for each year.

Now repeat the analysis when the interest rate is 2%

There’s a lot to summarize here.

When the interest rate was 3% we have the following. If you had sold the bond on any of the dates you would have the cash amounts given by the “total value,” shown in the 3% table.

When the interest rate was 2%, if you had sold the bond on any of the dates you would have the cash amounts given by the “total value,” shown in the 2% table.

The price and reinvestment effects at each date are shown by the difference between what you would have experienced when you first bought the bond at 3% and after the interest rate changed to 2%.

Since we are assuming the interest rate changed from 3% to 2%, you gain each period from the fact that your bond price has gone up (remember interest rates and bond prices move in opposite directions).

You lose on the reinvestment effect because you are only able to reinvest the coupons at 2% instead of the original 3% when you started.

The overall impact on your wealth is positive until the last year of the bond where you would be $2.49 worse off from owning the bond at 2% rather than 3%.

Watch the video below to see additional material on price and reinvestment risk.



Click here for: Worked Problems 12: Price risk and Reinvestment Risk

Click here for: In-Class Exercise 13: Price risk and Reinvestment Risk

Click here for: In-Class Exercise 14: Bond Concept Questions

2.6 Interactive Practice Questions: Introduction to Bonds