Debt Instruments 201


Debt Instruments 201

In this lesson, we will delve further into the characteristics and properties of bonds and become familiar with bond related terminology.  Bonds are often referred to as debt instruments, debt obligations, or fixed income securities (despite the fact that some bonds pay variable rates of interest, or none at all).

Coupon Rate

The rate of interest that a bond pays to the bondholder is called the coupon rate.  This term dates back to when bonds were issued as paper certificates (known as bearer bonds) with coupons attached.  Bearer bonds became the property of whoever had physical possession of the bond.  The bondholder would clip a coupon from the certificate and redeem it with the bond’s paying agent (usually a commercial bank) to collect their interest payment.  Now virtually all bonds are held in computerized book-entry form (all treasuries), or a physical certificate without coupons.

The coupon rate is the stated annualized interest rate that the bond issuer is contractually obligated to pay to the bondholder.  The coupon payment is usually paid on a quarterly, semi-annual, or annual basis and is based on the maturity date.  For example, if the maturity date is September 15, 2020, a bond that pays quarterly interest will make a payment on March 15, June 15, September 15, and December 15 each year until 2020.

The principal payment received at maturity and the basis for determining the amount of the coupon payment is the stated par amount over the life of the bond, regardless of the general level of interest rates. of the bond.  A bond that is paying a 7% coupon rate will pay $1,000 X .07, or $70 annually.  If the bond pays on a semi-annual basis, the bondholder will receive a $35 payment twice a year on the coupon payment date.  The coupon payment will not change (assuming the bond is not a floating-rate bond)

Zero Coupon Bonds

Zero coupon bonds do not make coupon payments during the life of the bond.  Zero coupon bonds are issued at a discount to par value, and the return to the investor comes in the difference that the bondholder pays for the bond (discount to par value), and the amount of principal payment made at maturity (par value).

Floating Rate Bonds

A floating rate bond has a coupon rate that is pegged to a benchmark, such as libor, and is adjusted periodically.  Floating rate bonds have the advantage of being less volatile in price, but the disadvantage of providing an unpredictable stream of income.  Floating-rate bonds are attractive to investors that are more interested in preservation of principal than in having an assured amount of income.

Par Amount

The par amount of a bond is the amount of principal that the bondholder will receive at maturity.  Par amount is also referred to as face amount.

Bonds are typically issued and traded in units of $1,000, though some issues have minimum amounts, such as $5,000.  The par amount of a bond being traded will determine how close to fair market price a particular bond will trade for.  The par amount that trades at the best market price is known as a round lot, and smaller sizes are known as odd lots.


The maturity dates is the specific day, month, and year that the bond issuer is obligated to pay back the principal to the bondholder.  There are rare examples of extendable bonds that give the issuer or bondholder the option to extend the maturity date of the bond.

In the bond market, the maturity of a bond is expressed as the number of years remaining until the bonds maturity date.  For example, a 30-year bond that was issued ten years ago would be referred to as a 20-year bond.

When bonds are issued, there is a direct correlation between the maturity date and the coupon rate of the bond.  The longer the length of time to maturity, the higher the coupon rate will be.  There are a number of reasons for this correlation; one of them is the time value of money. The time value of money stipulates that an individual would rather receive money now than at some time in the future due to the potential earnings capacity of the money.  Another reason for this is that the longer the time period, the more uncertainty is involved with holding the bond.  There is always a chance that the price of a bond can decline, or an adverse event can befall the issuer of the bond that will jeopardize their ability to pay the interest and/or principal.  Normally, this correlation also holds true for bonds trading in the secondary market, but there are rare occasion when it does not (we will discuss those occasions in later lessons).

Bond Pricing and Yield

As marketable debt instruments, bonds can be traded and their price can fluctuate over time.  Bond prices are quoted as a percentage of par amount.  For example, a bond price of 99 indicates a price of 99% of par, which would be $990 for a par amount of $1,000.  A bond trading at 101 3/8 is 101.375% of par or $1013.75 (every 1/8th of a point is worth $1.25 for every $1,000 in face amount).  Bonds priced above par (such as 102 ½) are referred to as trading at a premium, while bonds below par are at a discount. Bonds trading at 100 are always quoted as trading at par.

An entity that is offering bonds in the primary or new issue market, it sets the coupon rate to reflect the coupon rate of issues that are similar in maturity, credit quality, etc.  During the initial underwriting or offering period, the bonds should trade close to par, but when the offering period ends, the price can freely fluctuate on the secondary market.  We will examine what factors influence bond prices in detail in future lessons but they include:

  • The general level of interest rates;
  • The expected rate of inflation;
  • The economic outlook;
  • The issuer’s credit rating;
  • The supply of new debt issues; and
  • The demand for bonds versus alternative investments.

While we have stated that the coupon rate remains constant for fixed rate bonds, the market yield will fluctuate with the price of the bond.  If a bond has a coupon rate of 7 ¾% and the market yield for equivalent bond is 7 ¾%, then the price of the bond would be par.  But, if some event caused the market yield on equivalent bonds to rise to 8%, the coupon yield of the bond at 7 ¾% is not as attractive to potential buyers who can get a bond with the higher yield.  The price of the less desirable, lower yielding bond would drop in price until its market yield was also 8%.  This is a very important concept: When market yields increase, bond prices decrease; when market yields decrease, bond prices increase. That is to say that there is an inverse relationship between changes market yield and changes in bond prices.

When a bond is purchased at a price other than par, the interest payment is no longer the single factor that determines the rate of return on the investment.  If a bond is purchased at 90, the investor pays $900 for the bond, but will receive $1,000 at maturity.  In addition to receiving the interest payments over the life of the bond, the investor will also earn a $100 profit at maturity. Conversely, bonds that are purchased at a premium will cause the investor to experience a loss at maturity.  The measure of a bond’s return that takes this principal gain or loss at maturity is the bond’s yield to maturity.  The market yield that we have been referring to is the bond’s yield to maturity.  Yield to maturity is the measurement of the present value of a bond’s future cash flows (coupon payments and any principal gain or loss at maturity) based on the current market price.

We could go through the mathematics of calculating yield to maturity and how to determine what the price change of a bond would be for a given change in interest rates, but it is much easier to just use bond calculators, like those found here.  For those that are curious enough to want to explore the mathematics, you can find it here.

Another yield measurement that bond investors use is a bond’s current yield.  The current yield is determined by simply dividing the annual coupon payment by the market value of the bond.  A $1,000 par bond trading at 96 would have a market value of $960.  If the bond had a stated coupon rate of 7 ½% it would pay $75 annually:  75/960= .078125, so the current yield would be 7.8125%.  The current yield of a bond trading at a discount will always be higher than the coupon rate, while the current yield of a premium bond will always be less than the coupon rate.  This also holds true for yield to maturity.  Because yield to maturity takes into account the time value of money, and current yield does not, it is a much better measure of a bond’s true return.

Other Bond Properties

It is essential for anyone investing in bonds to carefully review all aspects of a bond before purchasing.  Here are some important properties of bands that should be carefully examined.

Credit Rating

There are three prominent independent credit rating services that rate fixed income securities- Moody’s, Standard and Poor’s (S&P), and Fitch.  These agencies assign letter ratings to bonds to grade their financial strength.  The top ratings are Aaa for Moody’s, and AAA for S&P and Fitch.  Here are the full spectrum of each agency’s rating grades:

Moody’s S&P Fitch
Aaa AAA AAA Prime
Aa1 AA+ AA+
Aa2 AA AA High Grade
Aa3 AA- AA-
A1 A+ A+
A2 A A Upper Medium Grade
A3 A- A-
Baa1 BBB+ BBB+
Baa2 BBB BBB Lower Medium Grade
Baa3 BBB- BBB-
Ba1 BB+ BB+ Non-Investment Grade
Ba2 BB BB Speculative
Ba3 BB- BB-
B1 B+ B+
B2 B B Highly Speculative
B3 B- B-
Caa1 CCC+ CCC Substantial Risks
Caa2 CCC In Poor Standing
Caa3 CCC-
Ca Extremely Speculative
C May Be In Default
D DD In Default

It is important to not only review the credit ratings, but to be conscious of changes in credit ratings.  A downgrading or threat of a downgrading will have an adverse effect on a bond’s price.  The agencies will often issue warnings of possible ratings downgrades through credit watches.  Particular caution should be given to Baa3/BBB-/BBB- bonds, as these are the lowest investment grade bonds in the market.  Many institutional investors are required to hold only investment grade bonds, so if these bonds get downgraded, some institutions will be forced to sell them, which will exacerbate the drop in price from the downgrading.

It is also important to keep an eye on bonds of competing firms in the same industry as the bond you are considering buying or already own.  Any ratings change that affects other bonds within the same sector can affect your bond.  For example, a downgrade of GM bonds can also affect the bonds of Ford Motors.

A bond’s yield relative to the market is largely determined by their credit rating.  The difference in yield between bonds of different ratings is known as the yield spread.  Yield spreads fluctuate along with economic conditions and expectations during the business cycle.  When the economy is weakening and there is fear of a recession, investors will sell lower rated bonds and buy higher rated bonds, particularly treasuries.  This is because financial weaker companies are less likely to have the resources to survive a severe decline in business.  Conversely, in the early stages of a recovery, investors will often be attracted to the high yields offered by lower rated bonds.  We will discuss yield spreads and related investment strategies in later lessons.

Embedded Options

Many bonds contain embedded options, such as a call option, sinking fund, put option, or the option to convert the bond into another security.  Bond investors need to be aware of any imbedded options, as they can have a profound effect on a bond’s value.

Callable Bonds

A call option gives the issuer the option to redeem a bond prior to maturity at a specified price.  A call option is an advantage for the issuer and a disadvantage to the investor.  Why is this?  Because an issuer will only call the bond when interest rates are low because they can issue new bonds at a lower rate of interest and reduce their interest expense.  The bondholder must then reinvest their funds at the lower current market rates.  Bonds that do not have call provisions are often called bullets.  Because of the early redemption risk associated with a call provision, callable bonds will yield more than a comparable bullet.  It is important to note that callable bonds will not appreciate as much as bullets when interest rates are declining because it is more likely that they will be called when rates are lower.

When considering investing in a callable bond, it is important to not only look at yield to maturity, but also consider the yield to call and yield to worst call.  Yield to call is calculated in the same way as yield to maturity, but using the earliest call date instead of the maturity date.  Because bonds are called when interest rates are low, it is prudent to assume that they will be called and the investor should be satisfied with the yield to call as well as yield to maturity when a bond is priced at a premium.  If the bond is priced at a discount, the yield to call will be higher than yield to maturity. and

Because call options give the issuer a series of future dates, it is important to calculate the yield to call for each date.  Sometimes different call dates will have different call prices.  The yield to call with the lowest return is considered the yield to worst call, and should be the return that the investor assumes they will receive.  A call schedule will specify the specific call dates and corresponding call price for each date.  Often, the call price will be a premium, such as 102, at the earliest call date and gradually decline to par over subsequent dates.  Bonds subject to a schedule are said to have discrete calls, while bonds that are callable at any time are known as continuous calls.

Again, it is not necessary to learn the mathematics involved, as you can use a yield to maturity calculator and substitute the call date and price.

Sinking Funds

A sinking fund provision provides for a retirement of portions of the bond issue periodically, often annually.  The bonds can be retired either by lottery, or on a pro-rata basis.  Like discrete call provisions, sinking fund dates and prices are listed on a schedule.  Some bonds may have both call and sinking fund provisions, and both need to be analyzed through yield to worst calculations.

Putable Bonds

Unlike call and sinking fund provisions, put options are favorable to the investor.  Buying a putable bond is like buying a bullet bond and an option.  The put provides a hedge against a rise in interest rates.  Because there is a price associated with a put, putable bonds will yield less than an otherwise equivalent bullet bond.  Often there is one, or a limited number of put dates on a bond.  Once the put date(s) pass, the bond becomes a bullet bond.

Putable bonds are analyzed by their yield to put in addition to yield to maturity.  Bonds with multiple put dates are analyzed by their yield to best put which is essentially the opposite of yield to worst call.

Duration and Convexity

By delving into duration and convexity, we are entering the realm of bond analysis that is often misunderstood and intimidating because of the mathematics involved.  Though the mathematics are not extremely complicated we will dispense with it because calculators are widely available to do the work for you, like the one found here, and it is much more important to understand their concepts and applications.

Macaulay Duration

The concept of duration dates back to 1938 when Frederick Macaulay began to regard the cash flow of a bond’s periodic interest payments and repayment of principal at maturity as a series of zero coupon bonds strung together.  Each individual payment was like a separate single-payment bond, and the bond was like a portfolio of these bonds.  Viewed in this light, the bond begins to mature at the first coupon payment.  The concept is easier to understand through an example:

Let’s assume we have bond with a $1,000 par amount that matures in 10 years and pays a 10% coupon rate. The bond pays $100 a year for ten years, a total of $1,000, and then pays $1,000 at maturity.  The total cash flow for the bond is $2,000.  After the first coupon is paid, the remaining cash flow is $1,900, and after the second it is $1,800, etc., until the final $1,000 principal payment at maturity.

Macaulay took each of the cash flows and multiplied them by their time period (the first was multiplied by 1, the next by 2, etc.).  He then divided the sum of these values by the current price of the bond and he called the result the bond’s duration.  We now refer to this as Macaulay duration, which is the average time to receipt of the cash flow, weighted by the present value of that cash flow, and it is stated in years.

If we perform this calculation on the bond in our example and assume it is priced to yield 9%, the Macaulay duration of the bond would be 6.86 years.  The Macaulay duration for a zero coupon bond would be its actual maturity, since this is the only cash flow that the bond pays.

Macaulay duration is most often used in a portfolio management strategy called immunization, which we will explore in a future lesson.  As a risk measure, modified duration is more useful.

Modified Duration and Convexity

Because bond prices move in the opposite direction to a change in interest rates, the risk of interest rates rising presents the greatest risk to a bondholder.   This is known as interest rate risk, and modified duration quantifies this risk by measuring how much a bond’s price will change with a given change in interest rates.

Modified duration is simply a measure of the percentage change in a bond’s price resulting from a specified change in market yield, typically 100 basis points, or 1%.   In other words, it measures a particular bond’s price volatility to a change in market yields.

A bond with a modified duration of 7.5 would experience a price change of 7.5% for each 100 basis point change in market yield.  If a bond is priced at $1,125.00 to yield 7% and the yield of equivalent bonds rose to 8%, the bonds price would drop by 7.5% to $1,068.75.  Conversely if rates dropped to 6%, the bonds price would rise to $1,181.25.

Interest rates do not always change by 100 basis points, but modified duration can be run through another equation to calculate a bond’s price change for any change in interest rates.  It is important to note that a bond’s duration is constantly changing.  This is because bond prices are constantly changing, which means that their yield is also constantly changing.  Because the duration is a function of price and yield, duration will also change.

As a bond approaches maturity, its price tends to approach par and this tendency is referred to as the bond’s pull to par.  The cause of this should be fairly obvious- because a bond will mature at par, no trader is going to pay a significant premium for a bond that will soon be worth par.  Conversely no trader will sell a bond at a large discount when maturity at par is eminent.  The concept of pull to par leads us to an important fundamental principle of duration: Longer maturity bonds experience more price movement than equivalent shorter maturity bonds for a given change in interest rates. To put it another way, longer maturity bonds have higher durations than shorter maturity bonds.

A bond’s coupon rate also affects a bond’s price sensitivity and, therefore, its duration: Lower coupon bonds experience more price movement than equivalent higher coupon bonds for a given change in interest rates. This makes perfect sense if you consider the cash flow structure of a bond, which is comprised of coupon payments and principal repayment.  Remember that a bond’s yield to maturity is determined by both the coupon payment and any loss (for a bond purchased at a premium) or gain (for a bond purchased at a discount) of principal at maturity.  Higher coupon bonds pay a greater portion of their yield in the form of interest, and less due to a gain or loss of principal, therefore lower coupon bonds will experience a larger price change for a given change in interest rates.  It may be helpful to consider the extreme example of zero coupon bonds.  Because a change in price from discount to par is the only return received from a zero coupon bond, it only stands to reason the changes in interest rates will have the most impact on the price of zero coupon bonds.

An understanding of duration leads us to the concept of convexity.  If we were to graph the relationship between the price and yield for a bond, the result would not be a straight line, but a curved, or convex, line like the one below.  The degree to which the line is curved shows how much a bond’s price changes as the result of a change in yield.

If we draw a straight line that touches the price/yield curve at a particular point, we have a tangent line to the price curve.  The slope of this line represents the bond’s duration.  For example a slope of 10 degrees indicates a bond’s price would change 10% for a 100 basis point change in yield.  However, this duration is only accurate for that specific point on the curve.  A tangent line drawn anywhere else on the curve would have a different slope.  If the price yield curve were a straight line, the price would always change by the same amount for a given in yield- the duration would be constant.  By virtue of the fact that the line is curved implies that the changes (duration) will occur in constantly changing amounts.  The rate of change of the duration is the convexity of the bond.  So duration measures the change in price for a given change in yield (and change in yield for a given change in price), and convexity measures the change in duration along the price/yield curve, and this change is asymmetrical.

Because of this asymmetrical relationship, duration is only good for predicting price changes for very small changes in yield.  For larger changes in yield, duration only provides an approximation of what the price change will be, and the bond’s convexity will also determine the price change.  The larger the change in yield, duration becomes less of a factor, and convexity becomes more of a factor.  Fortunately there are bond calculators available that take both factors in consideration and can predict price changes for large changes in interest rates.

Convexity can be used to compare two different bonds.  If two bonds have the same duration and yield, but one has greater convexity, that bond will be less affected by a change in interest rates than a bond with less convexity.  Also, bonds with greater convexity will have a higher price than bonds with a lower convexity regardless of any change in interest rates (see the diagram below).

Observe how both bonds have the same price and duration when price is P and yield is Y, but Bond A has greater convexity.  If interest rates change by a very small amount both bonds would still have about the same price.  But if there is a large change in interest rates (to Y’), Bond B’s price will decrease more than Bond A’s price.  Bond A’s higher price at Y’ shows that investors are willing to pay more for bonds with a higher convexity because they experience less price volatility when interest rates change; therefore they are subject to less interest rate risk.

If a bond is a bullet bond, the increase in price (and duration) as the yield decreases is greater than the decrease in price (and duration) as the yield increases (see the diagram below).

Notice how as the yield decreases, the slope of the duration increases.  This demonstrates that as yields decrease, the price will increase at an increasing rate.  Also notice how as yields increase, duration decreases, so prices decrease at a decreasing rate, making the bond less sensitive to price changes.  This shows that higher-yielding bonds are less price-sensitive to changing rates than lower-yielding bonds.  In other words, bonds are more sensitive to falling interest rates than rising interest rates.

If two bonds are identical except for the coupon rate, the bond with the lower rate will have higher convexity.  Therefore, zero-coupon bonds will have the highest convexity. Also, longer maturity bonds will have higher convexity than shorter maturity bonds.

So far we have limited our discussion to bullet bonds, but a call provision will have a profound impact on the convexity of a bond.  Below is a price/worst yield graph of a callable bond with a coupon rate Y.

As yields decrease below the coupon rate, the more likely it is that the bond will be called away.  As that becomes more likely the less investors are willing to pay above the call.  As the yield falls below the coupon rate the curve flattens out and the bond becomes negatively convex.  At yields above the coupon rate, the bond is essentially a bullet bond and has positive convexity.

While many investors avoid bonds with negative convexity, we will discuss situations in which negative convexity is beneficial to bond traders in a later lesson.

The following table summarizes how various factors influence a bonds price sensitivity to changes in interest rates:

More Price Sensitive Less Price Sensitive
Longer Maturity* Shorter Maturity*
Lower Coupon* Higher Coupon*
Higher Duration* Lower Duration*
Lower Convexity* Higher Convexity*

*All else being equal.


1 Comment

  1. […] an even greater differential of 59 basis points. Keep in mind that the farther out the maturity, the greater the sensitivity to changes in yield (aka duration) which in turn, increases the risk of the […]


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