Textbook Questions

The Behavior of Market Interest Rates

1. LG 1

2. LG 2

Recall from earlier discussions that rational investors try to earn a return that fully compensates them for risk. In the case of bondholders, that required return (ri) has three components: the real rate of return (r*), an expected inflation premium (IP), and a risk premium (RP). Thus, the required return on a bond can be expressed by the following equation:

ri=r*+IP+RPri=r*+IP+RPEquation11.1

The real rate of return and inflation premium are external economic factors, which together equal the risk-free rate (rf). To find the required return, we need to consider the unique features and properties of the bond issue itself that influence its risk. After we do this, we add a risk premium to the risk-free rate to obtain the required rate of return. A bond’s risk premium (RP) will take into account key issue and issuer characteristics, including such variables as the type of bond, the issue’s term to maturity, its call features, and its bond rating.

Together, the three components in  Equation 11.1  (r*, IP, and RP) drive the required return on a bond. Recall in the previous chapter that we identified five types of risks to which bonds are exposed. All of these risks are embedded in a bond’s required rate of return. That is, the bond’s risk premium addresses, among other things, the business and financial (credit) risk characteristics of an issue, along with its liquidity and call risks, whereas the risk-free rate (rf) takes into account interest rate and purchasing power risks.

Because these interest rates have a significant bearing on bond prices and yields, investors watch them closely. For example, more conservative investors watch interest rates because one of their major objectives is to lock in high yields. Aggressive traders also have a stake in interest rates because their investment programs are often built on the capital gains opportunities that accompany major swings in rates.

Keeping Tabs on Market Interest Rates

The bond market is not a single market. Rather, it consists of many different sectors. Similarly, there is no single interest rate that applies to all segments of the bond market. Instead, different interest rates apply to different segments. Granted, the various rates do tend to drift in the same direction over time, but it is also common for  yield spreads  (interest rate differentials) to exist among the various market sectors. Some important factors to keep in mind when you think about interest rates on bonds are as follows:

· Municipal bonds usually offer the lowest market rates because of their tax-exempt feature. As a rule, their market yields are about 20% to 30% lower than corporate bond yields.

· In the municipal sector, revenue bonds pay higher rates than general obligation bonds.

· In the taxable sector, Treasury securities have the lowest yields (because they have the least risk), followed by agency bonds and then corporate bonds, which provide the highest returns.

Famous Failures in Finance Signs of a Recession

When short-term interest rates on treasury bills exceed the rates on long-term treasury bonds, watch out. That is often the precursor to a recession. This “inversion” in the relationship between short-term and long-term rates has occurred prior to each of the last five U.S. recessions. Just as important, this indicator has rarely issued a false recession warning signal.

· Issues that normally carry bond ratings (e.g., municipals or corporates) generally display the same behavior: the lower the rating, the higher the yield.

· Most of the time, bonds with long maturities provide higher yields than short-term issues. However, this rule does not always hold. When short-term bond yields exceed yields on longer-term bonds, as they did in February 2006, that may be an early signal that a recession is coming.

· Bonds that are freely callable generally pay the highest interest rates, at least at date of issue. These are followed by deferred call obligations and then by noncallable bonds, which offer lower yields.

Watch Your Behavior

Anchoring on Credit Spreads The credit spread is the difference in yield between a risky bond and a safe bond. In theory, credit spreads are determined by forward-looking economic fundamentals that measure a borrower’s capacity to repay its debts. A recent study found that borrowers and lenders appear to focus excessively (i.e., to anchor) on past deal terms when setting spreads for a new bond issue. The study found that when a firm’s most recent past debt issue had a credit spread that was higher than an upcoming issue, the interest rate on the upcoming deal was higher than fundamentals could justify. In other words, both the firm and its lenders were anchored to the older, higher interest rate.

(Source: Casey Dougal, Joseph Engelberg, Christopher A. Parsons, & Edward D. Van Wesep, “Anchoring on Credit Spreads,” Journal of Finance, June 2015.)

As an investor, you should pay close attention to interest rates and yield spreads. Try to stay abreast of both the current state of the market and the future direction of market rates. Thus, if you are a conservative (income-oriented) investor and think that rates have just about peaked, that should be a signal to try to lock in the prevailing high yields with some form of call protection. (For example, buy bonds, such as Treasuries or AA-rated utilities that are noncallable or still have lengthy call deferments.) In contrast, if you’re an aggressive bond trader who thinks rates have peaked (and are about to drop), that should be a clue to buy bonds that offer maximum price appreciation potential (low-coupon bonds that still have a long time before they mature).

But how do you formulate such expectations? Unless you have considerable training in economics, you will probably need to rely on various published sources. Fortunately, a wealth of such information is available. Your broker is an excellent source for such reports, as are investor services like Moody’s and Standard & Poor’s. Also, of course, there are numerous online sources. Finally, there are widely circulated business and financial publications (like the Wall Street Journal, Forbes, Business Week, and Fortune) that regularly address the current state and future direction of market interest rates. Predicting the direction of interest rates is not easy. However, by taking the time to read some of these publications and reports regularly and carefully, you can at least get a sense of what experts predict is likely to occur in the near future.

What Causes Rates to Move?

Although the determination of interest rates is a complex economic issue, we do know that certain forces are especially important in influencing rate movements. Serious bond investors should make it a point to become familiar with the major determinants of interest rates and try to monitor those variables, at least informally.

Watch Your Behavior

Money Illusion An investment that offers a high interest rate may seem attractive, but remember it’s the real return, after inflation, that matters. Although interest rates were very high in the late 1970s, so was the inflation rate, and many bond investors earned negative real returns during that period.

In that regard, perhaps no variable is more important than inflation. Changes in the inflation rate, or to be more precise, changes in the expected inflation rate, have a direct and profound effect on market interest rates. When investors expect inflation to slow down, market interest rates generally fall as well. To gain an appreciation of the extent to which interest rates are linked to inflation, look at  Figure 11.1 . The figure plots the behavior of the interest rate on a 10-year U.S. Treasury bond and the inflation rate from 1963 to 2014. The blue line in the figure tracks the actual inflation rate over time, although as we have already noted, the expected inflation rate has a more direct effect on interest rates. Even so, there is a clear link between actual inflation and interest rates. Note that, in general, as inflation drifts up, so do interest rates. On the other hand, a decline in inflation is matched by a similar decline in interest rates. Most of the time, the rate on the 10-year bond exceeded the inflation rate, which is exactly what you should expect. When that was not the case, such as in the 1970s and more recently in 2012, investors in the 10-year Treasury bond did not earn enough interest to keep up with inflation. Notice that in 2009 as the U.S. struggled to recover from the Great Recession, the inflation rate was negative and the Treasury yields dropped sharply. On average, the 10-year Treasury yield exceeded the inflation rate by about 2.4 percentage points per year.

Figure 11.1 The Impact of Inflation on the Behavior of Interest Rates

The behavior of interest rates has always been closely tied to the movements in the rate of inflation. Since 1963 the average spread between the U.S. 10-year Treasury rate and inflation is 2.4 percentage points. This spread fluctuates quite a bit over time. Some extreme examples occurred in 1974 when the rate of inflation exceeded the 10-year Treasury rate by 4.1 percentage points and in 1985 when 10-year Treasury rates outpaced inflation by 8 percentage points.

In addition to inflation, five other important economic variables can significantly affect the level of interest rates:

· Changes in the money supply. An increase in the money supply pushes rates down (as it makes more funds available for loans), and vice versa. This is true only up to a point, however. If the growth in the money supply becomes excessive, it can lead to inflation, which, of course, means higher interest rates.

· The size of the federal budget deficit. When the U.S. Treasury has to borrow large amounts to cover the budget deficit, the increased demand for funds exerts an upward pressure on interest rates. That’s why bond market participants become so concerned when the budget deficit gets bigger and bigger—other things being equal, that means more upward pressure on market interest rates.

· The level of economic activity. Businesses need more capital when the economy expands. This need increases the demand for funds, and rates tend to rise. During a recession, economic activity contracts, and rates typically fall.

· Policies of the Federal Reserve. Actions of the Federal Reserve to control inflation also have a major effect on market interest rates. When the Fed wants to slow actual (or anticipated) inflation, it usually does so by driving up interest rates, as it did repeatedly in the mid- and late 1970s. Unfortunately, such actions sometimes have the side effect of slowing down business activity as well. Likewise, when the Federal Reserve wants to stimulate the economy, it takes action to push interest rates down, as it did repeatedly during and after the 2008-2009 recession.

· The level of interest rates in major foreign markets. Today investors look beyond national borders for investment opportunities. Rising rates in major foreign markets put pressure on rates in the United States to rise as well; if U.S. rates don’t keep pace, foreign investors may be tempted to dump their dollars to buy higher-yielding foreign securities.

Living Yield Curve

The Term Structure of Interest Rates and Yield Curves

Bonds having different maturities typically have different interest rates. The relationship between interest rates (yield) and time to maturity for any class of similar-risk securities is called the  term structure of interest rates . This relationship can be depicted graphically by a  yield curve , which shows the relation between time to maturity and yield to maturity for a group of bonds having similar risk. The yield curve constantly changes as market forces push bond yields at different maturities up and down.

Types of Yield Curves

Two types of yield curves are illustrated in  Figure 11.2 . By far, the most common type is curve 1, the red upward-sloping curve. It indicates that yields tend to increase with longer maturities. That’s partly because the longer a bond has to maturity, the greater the potential for price volatility. Investors, therefore, require higher-risk premiums to induce them to buy the longer, riskier bonds. Long-term rates may also exceed short-term rates if investors believe short-term rates will rise. In that case, rates on long-term bonds might have to be higher than short-term rates to attract investors. That is, if investors think short-term rates are rising, they will not want to tie up their money for long at today’s lower rates. Instead, they would prefer to invest in a short-term security so that they can reinvest that money quickly after rates have risen. To induce investors to purchase a long-term bond, the bond must offer a higher rate than investors think they could earn by buying a series of short-term bonds, with each new bond in that series offering a higher rate than the one before.

Figure 11.2 Two Types of Yield Curves

A yield curve plots the relation between term to maturity and yield to maturity for a series of bonds that are similar in terms of risk. Although yield curves come in many shapes and forms, the most common is the upward-sloping curve. It shows that yields increase with longer maturities.

Occasionally, the yield curve becomes inverted, or downward sloping, as shown in curve 2, which occurs when short-term rates are higher than long-term rates. This curve sometimes results from actions by the Federal Reserve to curtail inflation by driving short-term interest rates up. An inverted yield curve may also occur when firms are very hesitant to borrow long-term (such as when they expect a recession). With very low demand for long-term loans, long-term interest rates fall. In addition to these two common yield curves, two other types appear from time to time: the flat yield curve, when rates for short- and long-term debt are essentially the same, and the humped yield curve, when intermediate-term rates are the highest.

Plotting Your Own Curves

Yield curves are constructed by plotting the yields for a group of bonds that are similar in all respects but maturity. Treasury securities (bills, notes, and bonds) are typically used to construct yield curves. There are several reasons for this. Treasury securities have no risk of default. They are actively traded, so their prices and yields are easy to observe, and they are relatively homogeneous with regard to quality and other issue characteristics. Investors can also construct yield curves for other classes of debt securities, such as A-rated municipal bonds, Aa-rated corporate bonds, and even certificates of deposit.

Historical Yield Curves

Figure 11.3  shows the yield curves for Treasury securities on March 7, 2007, and March 16, 2015. To draw these curves, you need Treasury quotes from the U.S. Department of the Treasury or some other similar source. (Note that actual quoted yields for curve 1 are highlighted in yellow in the table below the graph.) Given the required quotes, select the yields for the Treasury bills, notes, and bonds maturing in approximately 1 month, 3 months, 6 months, and 1, 2, 3, 5, 7, 10, 20, and 30 years. That covers the full range of Treasury issues’ maturities. Next, plot the points on a graph whose horizontal (x) axis represents time to maturity in years and whose vertical (y) axis represents yield to maturity. Now, just connect the points to create the curves shown in  Figure 11.3 . You’ll notice that curve 1 is upward sloping, while curve 2 is downward sloping. Downward-sloping yield curves are less common,

 

Figure 11.3 Yield Curves on U.S. Treasury Issues

Here we see two yield curves constructed from actual market data obtained from the U.S. Department of the Treasury. Curve 2 shows a less common downward-sloping yield curve. The yields that make up the more common upward-sloping curve 1 are near U.S. record low levels.

(Source: U.S. Department of the Treasury, June 4, 2015.)

thankfully so because they often signal an upcoming recession. For example, the downward-sloping yield curve shown in  Figure 11.3  signaled the Great Recession that officially ran from December 2007 to June of 2009. While curve 1 is the more typical upward-sloping yield curve, it nonetheless reflects the historically low interest rates that prevailed as the U.S. economy recovered from a deep recession.

Explanations of the Term Structure of Interest Rates

As we noted earlier, the shape of the yield curve can change over time. Three commonly cited theories—the expectations hypothesis, the liquidity preference theory, and the market segmentation theory—explain more fully the reasons for the general shape of the yield curve.

Expectations Hypothesis

The  expectations hypothesis  suggests that the yield curve reflects investor expectations about the future behavior of interest rates. This theory argues that the relationship between short-term and long-term interest rates today reflects investors’ expectations about how interest rates will change in the future. When the yield curve slopes upward, and long-term rates are higher than short-term rates, the expectations hypothesis interprets this as a sign that investors expect short-term rates to rise. That’s why long-term bonds pay a premium compared to short-term bonds. People will not lock their money away in a long-term investment when they think interest rates are going to rise unless the rate on the long-term investment is higher than the current rate on short-term investments.

For example, suppose the current interest rate on a 1-year Treasury bill is 5%, and the current rate on a 2-year Treasury note is 6%. The expectations hypothesis says that this pattern of interest rates reveals that investors believe that the rate on a 1-year Treasury bill will go up to 7% next year. Why? That’s the rate that makes investors today indifferent between locking their money away for 2 years and earning 6% on the 2-year note versus investing in the 1-year T-bill today at 5% and then next year reinvesting the money from that instrument into another 1-year T-bill paying 7%.

Investment Strategy	(1) Rate Earned This Year	(2) Rate Earned Next Year	(3) Return over 2 Years [(1)+(2)][(1)+(2)]
Buy 2-year note today	6%	6%	12%
Buy 1-year T-bill, then reinvest in another T-bill next year	5%	7%	12%

Only if the rate on a 1-year T-bill rises from 5% this year to 7% next year will investors be indifferent between these 2 strategies. Thus, according to the expectations hypothesis, an upward-sloping yield curve means that investors expect interest rates to rise, and a downward-sloping yield curve means that investors expect interest rates to fall.

Example

Suppose the yield curve is inverted, and 1-year bonds offer a 5% yield while 2-year bonds pay a 4.5% yield. According to the expectations hypothesis, what do investors expect the 1-year bond yield to be 1 year from now? Remember that the expectations hypothesis says today’s short-term and long-term interest rates are set at a level which makes investors indifferent between short-term and long-term bonds, given their beliefs about where interest rates are headed. Therefore, to determine the expected 1-year bond yield next year, you must determine what return in the second year would make investors just as happy to buy two 1-year bonds as they are to buy one 2-year bond.

· Return on a 2-year bond = 4.5% + 4.5%Return on a 2-year bond = 4.5% + 4.5%

· Return on two 1-year bonds = 5.0% + xReturn on two 1-year bonds = 5.0% + x

The x in the second equation represents the expected rate on the 1-year bond next year. The top equation shows that an investor earns 9% over 2 years by purchasing a 2-year bond, so to achieve the same return on a series of two 1-year bonds, the return in the second year must be 4%.

Liquidity Preference Theory

More often than not, yield curves have an upward slope. The expectations hypothesis would interpret this as a sign that investors usually expect rates to rise. That seems somewhat illogical. Why would investors expect interest rates to rise more often than they expect rates to fall? Put differently, why would investors expect interest rates to trend up over time? There is certainly no historical pattern to lead one to hold that view. One explanation for the frequency of upward-sloping yield curves is the  liquidity preference theory . This theory states that long-term bond rates should be higher than short-term rates because of the added risks involved with the longer maturities. In other words, because of the risk differential between long- and short-term debt securities, rational investors will prefer the less risky, short-term obligations unless they can be motivated, via higher interest rates, to invest in longer-term bonds. Even if investors do not expect short-term rates to rise, long-term bonds will still have to offer higher yields to attract investors.

Actually, there are a number of reasons why rational investors should prefer short-term securities. To begin with, they are more liquid (more easily converted to cash) and less sensitive to changing market rates, which means there is less price volatility. For a given change in market rates, the prices of longer-term bonds will show considerably more movement than the prices of short-term bonds. In addition, just as investors tend to require a premium for tying up funds for longer periods, borrowers will also pay a premium in order to obtain long-term funds. Borrowers thus assure themselves that funds will be available, and they avoid having to roll over short-term debt at unknown and possibly unfavorable rates. All of these preferences explain why higher rates of interest should be associated with longer maturities and why it’s perfectly rational to expect upward-sloping yield curves.

Market Segmentation Theory

Another often-cited theory, the  market segmentation theory , suggests that the market for debt is segmented on the basis of the maturity preferences of different financial institutions and investors. According to this theory, the yield curve changes as the supply and demand for funds within each maturity segment determines its prevailing interest rate. The equilibrium between the financial institutions that supply the funds for short-term maturities (e.g., banks) and the borrowers of those short-term funds (e.g., businesses with seasonal loan requirements) establishes interest rates in the short-term markets. Similarly, the equilibrium between suppliers and demanders in such long-term markets as life insurance and real estate determines the prevailing long-term interest rates.

An Advisor’s Perspective

Ryan McKeown Senior VP-Financial Advisor, Wealth Enhancement Group

“I pay very close attention to the yield curve.”

MyFinanceLab

The shape of the yield curve can slope either upward or downward, as determined by the general relationship between rates in each market segment. When supply outstrips demand for short-term loans, short-term rates are relatively low. If, at the same time, the demand for long-term loans is higher than the available supply of funds, then long-term rates will move up, and the yield curve will have an upward slope. If supply and demand conditions are reversed—with excess demand for borrowing in the short-term market and an excess supply of funds in the long-term market—the yield curve could slope down.

Which Theory Is Right?

All three theories of the term structure have at least some merit in explaining the shape of the yield curve. These theories tell us that, at any time, the slope of the yield curve is affected by the interaction of (1) expectations regarding future interest rates, (2) liquidity preferences, and (3) the supply and demand conditions in the short- and long-term market segments. Upward-sloping yield curves result from expectations of rising interest rates, lender preferences for shorter-maturity loans, and a greater supply of short- than of long-term loans relative to the respective demand in each market segment. The opposite conditions lead to a downward-sloping yield curve.

More about the Yield Curve

Using the Yield Curve in Investment Decisions

Bond investors often use yield curves in making investment decisions. Analyzing the changes in yield curves provides investors with information about future interest rate movements, which in turn affect the prices and returns on different types of bonds. For example, if the entire yield curve begins to move upward, it usually means that inflation is starting to heat up or is expected to do so in the near future. In that case, investors can expect that interest rates, too, will rise. Under these conditions, most seasoned bond investors will turn to short or intermediate (three to five years) maturities, which provide reasonable returns and at the same time minimize exposure to capital loss when interest rates go up. A downward-sloping yield curve signals that rates have peaked and are about to fall and that the economy is slowing down.

Another factor to consider is the difference in yields on different maturities—the “steepness” of the curve. For example, a steep yield curve is one where long-term rates are much higher than short-term rates. This shape is often seen as an indication that the spread between long-term and short-term rates is about to fall, either because long-term rates will fall or short-term rates will rise. Steep yield curves are generally viewed as a bullish sign. For aggressive bond investors, they could be the signal to start moving into long-term securities. Flatter yield curves, on the other hand, sharply reduce the incentive for going long-term since the difference in yield between the 5- and 30-year maturities can be quite small. Under these conditions, investors would be well advised to just stick with the 5- to 10-year maturities, which will generate about the same yield as long bonds but without the risks.

Concepts in Review

Answers available at  http://www.pearsonhighered.com/smart

1. 11.1 Is there a single market rate of interest applicable to all segments of the bond market, or is there a series of market yields? Explain and note the investment implications of such a market environment.

2. 11.2 Explain why interest rates are important to both conservative and aggressive bond investors. What causes interest rates to move, and how can you monitor such movements?

3. 11.3 What is the term structure of interest rates and how is it related to the yield curve? What information is required to plot a yield curve? Describe an upward-sloping yield curve and explain what it has to say about the behavior of interest rates. Do the same for a flat yield curve.

4. 11.4 How might you, as a bond investor, use information about the term structure of interest rates and yield curves when making investment decisions?

The Pricing of Bonds

1. LG 3

No matter who the issuer is, what kind of bond it is, or whether it’s fully taxable or tax-free, all bonds are priced using similar principles. That is, all bonds (including notes with maturities of more than one year) are priced according to the present value of their future cash flow streams. Indeed, once the prevailing or expected market yield is known, the whole process becomes rather mechanical.

Market yields largely determine bond prices. That’s because in the marketplace, investors first decide what yield is appropriate for a particular bond, given its risk, and then they use that yield to find the bond’s price (or market value). As we saw earlier, the appropriate yield on a bond is a function of certain market and economic forces (e.g., the risk-free rate of return and inflation), as well as key issue and issuer characteristics (like years to maturity and the issue’s bond rating). Together these forces combine to form the required rate of return, which is the rate of return the investor would like to earn in order to justify an investment in a given fixed-income security. The required return defines the yield at which the bond should be trading and serves as the discount rate in the bond valuation process.

Investor Facts

Prices Go Up, Prices Go Down We all know that when market rates go up, bond prices go down (and vice versa). But bond prices don’t move up and down at the same speed because they don’t move in a straight line. Rather, the relationship between market yields and bond prices is convex, meaning bond prices will rise at an increasing rate when yields fall and decline at a decreasing rate when yields rise. That is, bond prices go up faster than they go down. This is known as positive convexity, and it’s a property of all noncallable bonds. Thus, for a given change in yield, you stand to make more money when prices go up than you’ll lose when prices move down!

The Basic Bond Valuation Model

Generally speaking, when you buy a bond you receive two distinct types of cash flow: (1) periodic interest income (i.e., coupon payments) and (2) the principal (or par value) at the end of the bond’s life. Thus, in valuing a bond, you’re dealing with an annuity of coupon payments for a specified number of periods plus a large single cash flow at maturity. You can use these cash flows, along with the required rate of return on the investment, in a present value-based bond valuation model to find the dollar value, or price, of a bond. Using annual compounding, you can calculate the price of a particular bond (BPi) using the following equation:

BPi=N∑t=1C(1+ri)t+PVN(1+ri)N=Present value of coupon payments + Present value of bond’s par valueBPi=∑t=1NC(1+ri)t+PVN(1+ri)N=Present value of coupon payments + Present value of bond’s par valueEquation11.2

where

· BPi = current price (or value) of a particular bond i

· C = annual coupon (interest) payment

· PVN = par value of the bond, at maturity

· N = number of years to maturity

· ri = prevailing market yield, or required annual return on bonds similar to bond i

In this form, you can compute the bond’s current value, or what you would be willing to pay for it, given that you want to generate a certain rate of return, as defined by ri. Alternatively, if you already know the bond’s price, you can solve for ri in the equation, in which case you’d be looking for the yield to maturity embedded in the current market price of the bond.

In the discussion that follows, we will demonstrate the bond valuation process in two ways. First, we’ll use annual compounding—that is, because of its computational simplicity, we’ll assume we are dealing with coupons that are paid once a year. Second, we’ll examine bond valuation under conditions of semiannual compounding, which is the way most bonds actually pay their interest.

Annual Compounding

You need the following information to value a bond: (1) the annual coupon payment, (2) the par value (usually $1,000), and (3) the number of years (i.e., time periods) remaining to maturity. You then use the prevailing market yield, ri, as the discount rate to compute the bond’s price, as follows:

Bond price=Present value of coupon payments + Present value ofbond’s par valueBond price = Present value of coupon payments + Present value of bond’s par valueEquation11.3

BPi=C(1+ri)1+C(1+ri)2+…+C(1+ri)N+$1,000(1+ri)NBPi=C(1+ri)1+C(1+ri)2+…+C(1+ri)N+$1,000(1+ri)NEquation11.3a

where again

· C = annual coupon payment

· N = number of years to maturity

Example

A 20-year, 4.5% bond is priced to yield 5%. That is, the bond pays an annual coupon of 4.5% (or $45), has 20 years left to maturity, and has a yield to maturity of 5%, which is the current market rate on bonds of this type. We can use  Equation 11.3  to find the bond’s price.

BPi=$45(1+0.05)1+$45(1+0.05)2+…+$45(1+0.05)20+$1,000(1+0.05)20=$937.69BPi=$45(1+0.05)1+$45(1+0.05)2+…+$45(1+0.05)20+$1,000(1+0.05)20=$937.69

Note that because this is a coupon-bearing bond, we have an annuity of coupon payments of $45 a year for 20 years, plus a single cash flow of $1,000 that occurs at the end of year 20. Thus, we find the present value of the coupon annuity and then add that amount to the present value of the recovery of principal at maturity. In this particular case, you should be willing to pay almost $938 for this bond, as long as you’re satisfied with earning 5% on your money.

Notice that this bond trades at a discount of $62.31 ($1,000 − $937.69)($1,000 − $937.69). It trades at a discount because its coupon rate (4.5%) is below the market’s required return (5%). You can directly link the size of the discount on this bond to the present value of the difference between the coupons that it pays ($45) and the coupons that would be required if the bond matched the market’s 5% required return ($50). In other words, this bond’s coupon payment is $5 less than what the market requires, so if you take the present value of that difference over the bond’s life, you will calculate the size of the bond’s discount:

$5(1+0.05)1+$5(1+0.05)2+…+$5(1+0.05)20=$62.31$5(1+0.05)1+$5(1+0.05)2+…+$5(1+0.05)20=$62.31

In a similar vein, for a bond that trades at a premium, the size of that premium equals the present value of the difference between the coupon that the bond pays and the (lower) coupon that the market requires.

Bonds initially sell for a price close to par value because bond issuers generally set the bond’s coupon rate equal or close to the market’s required return at the time the bonds are issued. If market interest rates change during the life of the bond, then the bond’s price will adjust up or down to reflect any differences between the bond’s coupon rate and the market interest rate. Although bonds can sell at premiums or discounts over their lives, as the maturity date arrives, bond prices will converge to par value. This happens because as time passes and a bond’s maturity date approaches, there are fewer interest payments remaining (so any premium or discount is diminishing) and the principal to be repaid at maturity is becoming an ever bigger portion of the bond’s price since the periods over which it is being discounted are disappearing.

Calculator Use

For annual compounding, to price a 20-year, 4.5% bond to yield 5%, use the keystrokes shown in the margin, where:

· N = number of years to maturity

· I = required annual return on the bond (what the bond is being priced to yield)

· PMT = annual coupon payment

· FV = par value of the bond

· PV = computed price of the bond

Recall that the calculator result shows the bond’s price as a negative value, which indicates that the price is a cash outflow for an investor when buying the bond’s cash flows.

Financial Calculator Tutorials

Spreadsheet Use

The bond’s price can also be calculated as shown on the following Excel spreadsheet.

Semiannual Compounding

Although using annual compounding simplifies the valuation process a bit, it’s not the way most bonds are actually valued in the marketplace. In practice, most bonds pay interest every six months, so it is appropriate to use semiannual compounding to value bonds. Fortunately, it’s relatively easy to go from annual to semiannual compounding: All you need to do is cut the annual interest income and the required rate of return in half and double the number of periods until maturity. In other words, rather than one compounding and payment interval per year, there are two (i.e., two 6-month periods per year). Given these changes, finding the price of a bond under conditions of semiannual compounding is much like pricing a bond using annual compounding. That is:

Bond price (with semiannual compounding)=Present value of the annuity ofsemiannual coupon payments+Present value of thebond’s par valueBond price (with semiannual compounding) = Present value of the annuity of semiannual coupon payments + Present value of the bond’s par valueEquation11.4

BPi=C/2(1+ri2)1+C/2(1+ri2)2+…+C/2(1+ri2)2N+$1,000(1+ri2)2NBPi=C/2(1+ri2)1+C/2(1+ri2)2+…+C/2(1+ri2)2N+$1,000(1+ri2)2NEquation11.4a

where, in this case,

· C/2 = semiannual coupon payment, or the amount of interest paid every 6 months

· ri = the required rate of return per 6-month period

Example

In the previous bond-pricing example, you priced a 20-year bond to yield 5%, assuming annual interest payments of $45. Suppose the bond makes semiannual interest payments instead. With semiannual payments of $22.50, you adjust the semiannual return to 2.5% and the number of periods to 40. Using  Equation 11.4 , you’d have:

BPi=$45/2(1+0.052)1+$45/2(1+0.052)2+…+$45/2(1+0.052)40+1,000(1+0.052)40=$937.24BPi=$45/2(1+0.052)1+$45/2(1+0.052)2+…+$45/2(1+0.052)40+1,000(1+0.052)40=$937.24

The price of the bond in this case ($937.24) is slightly less than the price we obtained with annual compounding ($937.69).

Calculator Use

For semiannual compounding, to price a 20-year, 4.5% semiannual-pay bond to yield 5%, use the keystrokes shown in the margin, where:

· N = number of 6-month periods to maturity (20×2=40)(20×2=40)

· I = yield on the bond, adjusted for semiannual compounding (5%÷2=2.5%)(5%÷2=2.5%)

· PMT = semiannual coupon payment ($45.00÷2=$22.50)($45.00÷2=$22.50)

· FV = par value of the bond

· PV = computed price of the bond

Spreadsheet Use

You can calculate the bond’s price with semiannual coupon payments as shown on the following Excel spreadsheet. Notice that in cell B8 the required annual return is divided by coupon payment frequency to find the required rate of return per 6-month period, and the number of years to maturity is multiplied times the coupon payment frequency to find the total number of 6-month periods remaining until maturity.

Accrued Interest

Most bonds pay interest every six months, but you can trade them any time that the market is open. Suppose you own a bond that makes interest payments on January 15 and July 15 each year. What happens if you sell this bond at some time between the scheduled coupon payment dates? For example, suppose you sell the bond on October 15, a date that is roughly halfway between two payment dates. Fortunately, interest accrues on bonds between coupon payments, so selling the bond prior to a coupon payment does not mean that you sacrifice any interest that you earned.  Accrued interest  is the amount of interest earned on a bond since the last coupon payment. When you sell a bond in between coupon dates, the bond buyer adds accrued interest to the bond’s price (the price calculated using  Equation 11.3  or  11.4  depending on whether coupons arrive annually or semiannually).

Example

Suppose you purchase a $1,000 par value bond that pays a 6% coupon in semiannual installments of $30. You received a coupon payment two months ago, and now you are ready to sell the bond. Contacting a broker, you learn that the bond’s current market price is $1,010. If you sell the bond, you will receive not only the market price, but also accrued interest. Because you are about one-third of the way between the last coupon payment and the next one, you receive accrued interest of $10 (i.e., 1/3 × $30)$10 (i.e., 1/3 × $30), so the total cash that you receive in exchange for your bond is $1,020.

Traders in the bond market sometimes refer to the price of a bond as being either clean or dirty. The  clean price  of a bond equals the present value of its cash flows, as in  Equations 11.3  and  11.4 . As a matter of practice, bond price quotations that you may find in financial periodicals or online are nearly always clean prices. The  dirty price  of a bond is the clean price plus accrued interest. In the example above, the clean price is $1,010, and the dirty price is $1,020.

Concepts in Review

Answers available at  http://www.pearsonhighered.com/smart

1. 11.5 Explain how market yield affects the price of a bond. Could you price a bond without knowing its market yield? Explain.

2. 11.6 Why are bonds generally priced using semiannual compounding? Does it make much difference if you use annual compounding?

Measures of Yield and Return

1. LG 4

In the bond market, investors focus as much on a bond’s yield to maturity as on its price. As you have seen, the yield to maturity helps determine the price at which a bond trades, but it also measures the rate of return on the bond. When you can observe the price of a bond that is trading in the market, you can simply reverse the bond valuation process described above to solve for the bond’s yield to maturity rather than its price. That gives you a pretty good idea of the return that you might earn if you purchased the bond at its current market price. Actually, there are three widely used metrics to assess the return on a bond: the current yield, the yield to maturity, and the yield to call (for bonds that are callable). We’ll look at all three measures here, along with a concept known as the expected return, which measures the expected (or actual) rate of return earned over a specific holding period.

Current Yield

The  current yield  is the simplest of all bond return measures, but it also has the most limited application. This measure looks at just one source of return: a bond’s annual interest income. In particular, it indicates the amount of current income a bond provides relative to its prevailing market price. The current yield equals:

Current yield=Annual interest incomeCurrent market price of the bondCurrent yield = Annual interest incomeCurrent market price of the bondEquation11.5

Example

An 8% bond would pay $80 per year in interest for every $1,000 of principal. However, if the bond was currently priced at $800, it would have a current yield of $80÷$800 = 0.10 = 10%$80÷$800 = 0.10 = 10%. The current yield measures a bond’s annual interest income, so it is of interest primarily to investors seeking high levels of current income, such as endowments or retirees.

Yield to Maturity

The  yield to maturity (YTM)  is the most important and most widely used measure of the return provided by a bond. It evaluates the bond’s interest income and any gain or loss that results from differences between the price that an investor pays for a bond and the par value that the investor receives at maturity. The YTM takes into account all of the cash flow received over a bond’s life. Also known as the  promised yield , the YTM shows the rate of return earned by an investor, given that the bond is held to maturity and all principal and interest payments are made in a prompt and timely fashion. In addition, the YTM calculation implicitly assumes that the investor can reinvest all the coupon payments at an interest rate equal to the bond’s yield to maturity. This “reinvestment assumption” plays a vital role in the YTM, which we will discuss in more detail later in this chapter (see the section entitled Yield Properties).

The yield to maturity is used not only to gauge the return on a single issue but also to track the behavior of the market in general. In other words, market interest rates are basically a reflection of the average promised yields that exist in a given segment of the market. The yield to maturity provides valuable insights into an issue’s investment merits that investors can use to assess the attractiveness of different bonds. Other things being equal, the higher the promised yield of an issue, the more attractive it is.

Although there are a couple of ways to compute the YTM, the best and most accurate procedure is derived directly from the bond valuation model described above. That is, you can use  Equations 11.3  and  11.4  to determine the YTM for a bond. The difference is that now instead of trying to determine the price of the bond, you know its price and are trying to find the discount rate that will equate the present value of the bond’s cash flow (coupon and principal payments) to its current market price. This procedure may sound familiar. It’s just like the internal rate of return measure described earlier in the text. Indeed, the YTM is basically the internal rate of return on a bond. When you find that, you have the bond’s yield to maturity.

Using Annual Compounding

Finding yield to maturity is a matter of trial and error. In other words, you try different values for YTM until you find the one that solves the equation. Let’s say you want to find the YTM for a 7.5% ($1,000 par value) annual-coupon-paying bond that has 15 years remaining to maturity and is currently trading in the market at $809.50. From  Equation 11.3 , we know that

BPi=$809.50=$75(1+ri)1+$75(1+ri)2+…+$75(1+ri)15+$1,000(1+ri)15BPi=$809.50=$75(1+ri)1+$75(1+ri)2+…+$75(1+ri)15+$1,000(1+ri)15

Notice that this bond sells below par (i.e., it sells at a discount). What do we know about the relationship between the required return on a bond and its coupon rate when the bond sells at a discount? Bonds sell at a discount when the required return (or yield to maturity) is higher than the coupon rate, so the yield to maturity on this bond must be higher than 7.5%.

Through trial and error, we might initially try a discount rate of 8% or 9% (or, since it sells at a discount, any value above the bond’s coupon). Sooner or later, we’ll try a discount rate of 10%, and at that discount rate, the present value of the bond’s cash flows is $809.85 (use  Equation 11.3  to verify this), which is very close to the bond’s market price.

Because the computed price of $809.85 is reasonably close to the bond’s current market price of $809.50, we can say that 10% represents the approximate yield to maturity on this bond. That is, 10% is the discount rate that leads to a computed bond price that’s equal (or very close) to the bond’s current market price. In this case, if you were to pay $809.50 for the bond and hold it to maturity, you would expect to earn a YTM very close to 10.0%. Doing trial and error by hand can be time consuming, so you can use a handheld calculator or computer software to calculate the YTM.

Calculator Use

For annual compounding, to find the YTM of a 15-year, 7.5% bond that is currently priced in the market at $809.50, use the keystrokes shown in the margin. The present value (PV) key represents the current market price of the bond, and all other keystrokes are as defined earlier.

Spreadsheet Use

The bond’s YTM can also be calculated as shown on the following Excel spreadsheet.

Using Semiannual Compounding

Given some fairly simple modifications, it’s also possible to find the YTM using semiannual compounding. To do so, we cut the annual coupon and discount rate in half and double the number of periods to maturity. Returning to the 7.5%, 15-year bond, let’s see what happens when you use  Equation 11.4  and try an initial discount rate of 10%.

BPi=$75.00/2(1+0.102)1$75.00/2(1+0.102)2+…+$75.00/2(1+0.102)30+$1,000(1+0.102)30=$807.85BPi=$75.00/2(1+0.102)1$75.00/2(1+0.102)2+…+$75.00/2(1+0.102)30+$1,000(1+0.102)30=$807.85

As you can see, a semiannual discount rate of 5% results in a computed bond value that’s well short of the market price of $809.50. Given the inverse relationship between price and yield, it follows that if you need a higher price, you have to try a lower YTM (discount rate). Therefore, you know the semiannual yield on this bond has to be something less than 5%. By trial and error, you would determine that the yield to maturity on this bond is just a shade under 5% per half year—approximately 4.99%. Remember that this is the yield expressed over a 6-month period. The market convention is to simply state the annual yield as twice the semiannual yield. This practice produces what the market refers to as the  bond equivalent yield . Returning to the YTM problem started above, you know that the issue has a semiannual yield of about 4.99%. According to the bond equivalent yield convention, you double the semiannual rate to obtain the annual rate of return on this bond. Doing this results in an annualized yield to maturity (or promised yield) of approximately 4.99%×2=9.98%4.99%×2=9.98%. This is the annual rate of return you will earn on the bond if you hold it to maturity.

Calculator Use

For semiannual compounding, to find the YTM of a 15-year, 7.5% bond that is currently priced in the market at $809.50, use the keystrokes shown here. As before, the PV key is the current market price of the bond, and all other keystrokes are as defined earlier. Remember that to find the bond equivalent yield, you must double the computed value of I, 4.987%. That is 4.987%×2=9.97%4.987%×2=9.97%. The difference between our answer here, 9.97%, and the 9.98% figure in the previous paragraph is simply due to the calculator’s more precise rounding.

Spreadsheet Use

A semiannual bond’s YTM and bond equivalent yield can also be calculated as shown on the following Excel spreadsheet.

Yield Properties

Actually, in addition to holding the bond to maturity, there are several other critical assumptions embedded in any yield to maturity figure. The promised yield measure—whether computed with annual or semiannual compounding—is based on present value concepts and therefore contains important reinvestment assumptions. To be specific, the YTM calculation assumes that when each coupon payment arrives, you can reinvest it for the remainder of the bond’s life at a rate that is equal to the YTM. When this assumption holds, the return that you earn over a bond’s life is in fact equal to the YTM. In essence, the calculated yield to maturity figure is the return “promised” only as long as the issuer meets all interest and principal obligations on a timely basis and the investor reinvests all interest income at a rate equal to the computed promised yield. In our example above, you would need to reinvest each of the coupon payments and earn a 10% return on those reinvested funds. Failure to do so would result in a realized yield of less than the 10% YTM. If you made no attempt to reinvest the coupons, you would earn a realized yield over the 15-year investment horizon of just over 6.5%—far short of the 10% promised return. On the other hand, if you could reinvest coupons at a rate that exceeded 10%, the actual yield on your bond over the 15 years would be higher than its 10% YTM. The bottom line is that unless you are dealing with a zero-coupon bond, a significant portion of the bond’s total return over time comes from reinvested coupons.

When we use present value-based measures of return, such as the YTM, there are actually three components of return: (1) coupon/interest income, (2) capital gains (or losses), and (3) interest on interest. Whereas current income and capital gains make up the profits from an investment, interest on interest is a measure of what you do with those profits. In the context of a bond’s yield to maturity, the computed YTM defines the required, or minimum, reinvestment rate. Put your investment profits (i.e., interest income) to work at this rate and you’ll earn a rate of return equal to YTM. This rule applies to any coupon-bearing bond—as long as there’s an annual or semiannual flow of interest income, the reinvestment of that income and interest on interest are matters that you must deal with. Also, keep in mind that the bigger the coupon and/or the longer the maturity, the more important the reinvestment assumption. Indeed, for many long-term, high-coupon bond investments, interest on interest alone can account for well over half the cash flow.

Finding the Yield on a Zero

You can also use the procedures described above ( Equation 11.3  with annual compounding or  Equation 11.4  with semiannual compounding) to find the yield to maturity on a zero-coupon bond. The only difference is that you can ignore the coupon portion of the equation because it will, of course, equal zero. All you need to do to find the promised yield on a zero-coupon bond is to solve the following expression:

Yield=($1,000Price)1N−1Yield = ($1,000Price)1N−1

Example

Suppose that today you could buy a 15-year zero-coupon bond for $315. If you purchase the bond at that price and hold it to maturity, what is your YTM?

Yield=($1,000$315)115−1=0.08=8%Yield = ($1,000$315)115−1=0.08=8%

The zero-coupon bond pays an annual compound return of 8%. Had we been using semiannual compounding, we’d use the same equation except we’d substitute 30 for 15 (because there are 30 semiannual periods in 15 years). The yield would change to 3.93% per half year, or 7.86% per year.

Calculator Use

For semiannual compounding, to find the YTM of a 15-year zero-coupon bond that is currently priced in the market at $315, use the keystrokes shown in the margin. PV is the current market price of the bond, and all other keystrokes are as defined earlier. To find the bond equivalent yield, double the computed value of I, 3.926%. That is, 3.926%×2 = 7.85%3.926%×2 = 7.85%.

Spreadsheet Use

A semiannual bond’s YTM and bond equivalent yield can also be calculated as shown on the following Excel spreadsheet. Notice that the spreadsheet also shows 7.85% for the bond equivalent yield.

Yield to Call

Bonds can be either noncallable or callable. Recall that a noncallable bond prohibits the issuer from calling the bond prior to maturity. Because such issues will remain outstanding to maturity, you can value them by using the standard yield to maturity measure. In contrast, a callable bond gives the issuer the right to retire the bond before its maturity date, so the issue may not remain outstanding to maturity. As a result, the YTM may not always provide a good measure of the return that you can expect if you purchase a callable bond. Instead, you should consider the impact of the bond being called away prior to maturity. A common way to do that is to use a measure known as the  yield to call (YTC) , which shows the yield on a bond if the issue remains outstanding not to maturity but rather until its first (or some other specified) call date.

The YTC is commonly used with bonds that carry deferred-call provisions. Remember that such issues start out as noncallable bonds and then, after a call deferment period (of 5 to 10 years), become freely callable. Under these conditions, the YTC would measure the expected yield on a deferred-call bond assuming that the issue is retired at the end of the call deferment period (that is, when the bond first becomes freely callable). You can find the YTC by making two simple modifications to the standard YTM equation ( Equation 11.3  or  11.4 ). First, define the length of the investment horizon (N) as the number of years to the first call date, not the number of years to maturity. Second, instead of using the bond’s par value ($1,000), use the bond’s call price (which is stated in the indenture and is frequently greater than the bond’s par value).

For example, assume you want to find the YTC on a 20-year, 10.5% deferred-call bond that is currently trading in the market at $1,204 but has five years to go to first call (that is, before it becomes freely callable), at which time it can be called in at a price of $1,085. Rather than using the bond’s maturity of 20 years in the valuation equation ( Equation 11.3  or  11.4 ), you use the number of years to first call (five years), and rather than the bond’s par value, $1,000, you use the issue’s call price, $1,085. Note, however, you still use the bond’s coupon (10.5%) and its current market price ($1,204). Thus, for annual compounding, you would have:

BPi=$1,204=$105(1+ri)1+$105(1+ri)2+$105(1+ri)3+$105(1+ri)4+$105(1+ri)5+$1,085(1+ri)5BPi=$1,204=$105(1+ri)1+$105(1+ri)2+$105(1+ri)3+$105(1+ri)4+$105(1+ri)5+$1,085(1+ri)5Equation11.6

Through trial and error, you could determine that at a discount rate of 7%, the present value of the future cash flows (coupons over the next five years, plus call price) will exactly (or very nearly) equal the bond’s current market price of $1,204.

Thus, the YTC on this bond is 7%. In contrast, the bond’s YTM is 8.37%. In practice, bond investors normally compute both YTM and YTC for deferred-call bonds that are trading at a premium. They do this to find which yield is lower; the market convention is to use the lower, more conservative measure of yield (YTM or YTC) as the appropriate indicator of the bond’s return. As a result, the premium bond in our example would be valued relative to its yield to call. The assumption is that because interest rates have dropped so much (the YTM is two percentage points below the coupon rate), it will be called in the first chance the issuer gets. However, the situation is totally different when this or any bond trades at a discount. Why? Because the YTM on any discount bond, whether callable or not, will always be less than the YTC. Thus, the YTC is a totally irrelevant measure for discount bonds—it’s used only with premium bonds.

Calculator Use

For annual compounding, to find the YTC of a 20-year, 10.5% bond that is currently trading at $1,204 but can be called in five years at a call price of $1,085, use the keystrokes shown in the margin. In this computation, N is the number of years to first call date, and FV represents the bond’s call price. All other keystrokes are as defined earlier.

Spreadsheet Use

A callable bond’s YTC can also be calculated as shown on the following Excel spreadsheet.

Expected Return

Rather than just buying and holding bonds, some investors prefer to actively trade in and out of these securities over fairly short investment horizons. As a result, measures such as yield to maturity and yield to call have relatively little meaning, other than as indicators of the rate of return used to price the bond. These investors obviously need an alternative measure of return that they can use to assess the investment appeal of those bonds they intend to trade. Such an alternative measure is the  expected return . It indicates the rate of return an investor can expect to earn by holding a bond over a period of time that’s less than the life of the issue. (Expected return is also known as  realized yield  because it shows the return an investor would realize by trading in and out of bonds over short holding periods.)

The expected return lacks the precision of the yield to maturity (and YTC) because the major cash flow variables are largely the product of investor estimates. In particular, going into the investment, both the length of the holding period and the future selling price of the bond are pure estimates and therefore subject to uncertainty. Even so, you can use essentially the same procedure to find a bond’s realized yield as you did to find the promised yield. That is, with some simple modifications to the standard bond-pricing formula, you can use the following equation to find the expected return on a bond.

Bond price=Present value of the bond'sannual coupon paymentsover the holding period+Present value of the bond'sfuture price at the endof the holding periodBond price = Present value of the bond's annual coupon payments over the holding period+Present value of the bond's future price at the end of the holding periodEquation11.7

BPi=C(1+ri)1+C(1+ri)2+…+C(1+ri)N+FV(1+ri)NBPi=C(1+ri)1+C(1+ri)2+…+C(1+ri)N+FV(1+ri)NEquation11.7a

where this time N represents the length of the holding period (not years to maturity), and FV is the expected future price of the bond.

As indicated above, you must determine the future price of the bond when computing its expected return. This is done by using the standard bond price formula, as described earlier. The most difficult part of deriving a reliable future price is, of course, coming up with future market interest rates that you feel will exist when the bond is sold. By evaluating current and expected market interest rate conditions, you can estimate the YTM that you expect the issue to provide at the date of sale and then use that yield to calculate the bond’s future price.

To illustrate, take one more look at our 7.5%, 15-year bond. This time, let’s assume that you feel the price of the bond, which is now trading at a discount, will rise sharply as interest rates fall over the next few years. In particular, assume the bond is currently priced at $809.50 (to yield 10%) and you anticipate holding the bond for three years. Over that time, you expect market rates to drop to 8%. With that assumption in place, and recognizing that three years from now the bond will have 12 remaining coupon payments, you can use  Equation 11.3  to estimate that the bond’s price will be approximately $960 in three years. Thus, you are assuming that you will buy the bond today at a market price of $809.50 and sell it three years later—after interest rates have declined to 8%—at a price of $960. Given these assumptions, the expected return (realized yield) on this bond is 14.6%, which is the discount rate in the following equation that will produce a current market price of $809.50.

BPi=$809.50=$75(1+ri)1+$75(1+ri)2+$75(1+ri)3+$960(1+ri)3BPi=$809.50=$75(1+ri)1+$75(1+ri)2+$75(1+ri)3+$960(1+ri)3

where

ri=0.146=14.6%.ri=0.146=14.6%.

The return on this investment is fairly substantial, but keep in mind that this is only an estimate. It is, of course, subject to variation if things do not turn out as anticipated, particularly with regard to the market yield expected at the end of the holding period. This example uses annual compounding, but you could just as easily have used semiannual compounding, which, everything else being the same, would have resulted in an expected yield of 14.4% rather than the 14.6% found with annual compounding.

Calculator Use

For semiannual compounding, to find the expected return on a 7.5% bond that is currently priced in the market at $809.50 but is expected to rise to $960 within a three-year holding period, use the keystrokes shown in the margin. In this computation, PV is the current price of the bond, and FV is the expected price of the bond at the end of the (three-year) holding period. All other keystrokes are as defined earlier. To find the bond equivalent yield, double the computed value of I, 7.217%. That is 7.217%×2=14.43%7.217%×2=14.43%.

Spreadsheet Use

The expected return for semiannual compounding can also be calculated as shown on the following Excel spreadsheet. Notice that the spreadsheet shows 14.43% for the bond equivalent yield.

Valuing a Bond

Depending on their objectives, investors can estimate the return that they will earn on a bond by calculating either its yield to maturity or its expected return. Conservative, income-oriented investors focus on the YTM. Earning interest income over extended periods of time is their primary objective, above earning a quick capital gain if interest rates fall. Because these investors intend to hold most of the bonds that they buy to maturity, the YTM (or the YTC) is a reliable measure of the returns that they can expect over time—assuming, of course, the reinvestment assumptions embedded in the yield measure are reasonable. More aggressive bond traders, who hope to profit from swings in market interest rates, calculate the expected return to estimate the return that they will earn on a bond. Earning capital gains by purchasing and selling bonds over relatively short holding periods is their chief concern, so the expected return is more important to them than the YTM.

In either case, the promised or expected yield provides a measure of return that investors can use to determine the relative attractiveness of fixed-income securities. But to evaluate the merits of different bonds, we must evaluate their returns and their risks. Bonds are no different from stocks in that the return (promised or expected) that they provide should be sufficient to compensate investors for the risks that they take. Thus, the greater the risk, the greater the return the bond should generate.

Concepts in Review

Answers available at  http://www.pearsonhighered.com/smart

1. 11.7 What’s the difference between current yield and yield to maturity? Between promised yield and realized yield? How does YTC differ from YTM?

2. 11.8 Briefly describe the term bond equivalent yield. Is there any difference between promised yield and bond equivalent yield? Explain.

3. 11.9 Why is the reinvestment of interest income so important to bond investors?

Duration and Immunization

1. LG 5

One of the problems with the yield to maturity is that it assumes you can reinvest the bond’s periodic coupon payments at the same rate over time. If you reinvest this interest income at a lower rate (or if you spend it), your actual return will be lower than the YTM. Another flaw is that YTM assumes the investor will hold the bond to maturity. If you sell a bond prior to its maturity, the price that you receive will reflect prevailing interest rates, which means that the return that you will earn will probably differ from the YTM. If rates have moved up since you purchased the bond, the bond will sell at a discount, and your return will be less than the YTM. If interest rates have dropped, the opposite will happen.

The problem with yield to maturity, then, is that it fails to take into account the effects of reinvestment risk and price (or market) risk. To see how reinvestment and price risks behave relative to one another, consider a situation in which market interest rates have undergone a sharp decline. Under such conditions, bond prices will rise. You might be tempted to cash out your holdings and take some gains (i.e., do a little “profit taking”). Indeed, selling before maturity is the only way to take advantage of falling interest rates because a bond will pay its par value at maturity, regardless of prevailing interest rates. That’s the good news about falling rates, but there is a downside. When interest rates fall, so do the opportunities to reinvest at high rates. Therefore, although you gain on the price side, you lose on the reinvestment side. Even if you don’t sell out, you are faced with decreased reinvestment opportunities. To earn the YTM promised on your bonds, you must reinvest each coupon payment at the same YTM rate. Obviously, as rates fall, you’ll find it increasingly difficult to reinvest the stream of coupon payments at that rate. When market rates rise, just the opposite happens. The price of the bond falls, but your reinvestment opportunities improve.

Bond investors need a measure that helps them judge just how significant these risks are for a particular bond. Such a yardstick is provided by something called  duration . It captures in a single measure the extent to which the price of a bond will react to different interest rate environments. Because duration gauges the price volatility of a bond, it gives you a better idea of how likely you are to earn the return (YTM) you expect. That, in turn, will help you tailor your holdings to your expectations of interest rate movements.

The Concept of Duration

The concept of duration was first developed in 1938 by actuary Frederick Macaulay to help insurance companies match their cash inflows with payments. When applied to bonds, duration recognizes that the amount and frequency of interest payments, the yield to maturity, and the term to maturity all affect the interest rate risk of a particular bond. Term to maturity is important because it influences how much a bond’s price will rise or fall as interest rates change. In general, when rates move, bonds with longer maturities fluctuate more than shorter issues. On the other hand, while the amount of price risk embedded in a bond is related to the issue’s term to maturity, the amount of reinvestment risk is directly related to the size of a bond’s coupon. Bonds that pay high coupons have greater reinvestment risk simply because there’s more to reinvest.

As it turns out, both price and reinvestment risk are related in one way or another to interest rates, and therein lies the conflict. Any change in interest rates (whether up or down) will cause price risk and reinvestment risk to push and pull bonds in opposite directions. An increase in rates will produce a drop in price but will increase reinvestment opportunities. Declining rates, in contrast, will boost prices but decrease reinvestment opportunities. At some point in time, these two forces should exactly offset each other. That point in time is a bond’s duration.

In general, bond duration possesses the following properties:

· Higher coupons result in shorter durations.

· Longer maturities mean longer durations.

· Higher yields (YTMs) lead to shorter durations.

Together these variables—coupon, maturity, and yield—interact to determine an issue’s duration. Knowing a bond’s duration is helpful because it captures the bond’s underlying price volatility. That is, since a bond’s duration and volatility are directly related, it follows that the shorter the duration, the less volatility in bond prices—and vice versa, of course.

Measuring Duration

Duration is a measure of the average maturity of a fixed-income security. The term average maturity may be confusing because bonds have only one final maturity date. An alternative definition of average maturity might be that it captures the average timing of the bond’s cash payments. For a zero-coupon bond that makes only one cash payment on the final maturity date, the bond’s duration equals its maturity. But because coupon-paying bonds make periodic interest payments, the average timing of these payments (i.e., the average maturity) is different from the actual maturity date. For instance, a 10-year bond that pays a 5% coupon each year distributes a small cash flow in year 1, in year 2, and so on up until the last and largest cash flow in year 10. Duration is a measure that puts some weight on these intermediate payments, so that the “average maturity” is a little less than 10 years.

You can think of duration as the weighted-average life of a bond, where the weights are the fractions of the bond’s total value accounted for by each cash payment that the bond makes over its life. Mathematically, we can find the duration of a bond as follows:

Duration=N∑t=1[PV(Ct)BP×t]Duration = ∑t=1N[PV(Ct)BP×t]Equation11.8

where

· PV(Ct) = present value of a future coupon or principal payment

· BP = current market price of the bond

· t = year in which the cash flow (coupon or principal) payment is received

· N = number of years to maturity

The duration measure obtained from  Equation 11.8  is commonly referred to as Macaulay duration—named after the actuary who developed the concept.

Although duration is often computed using semiannual compounding,  Equation 11.8  uses annual coupons and annual compounding to keep the ensuing discussion and calculations as simple as possible. Even so, the formula looks more formidable than it really is. If you follow the basic steps noted below, you’ll find that duration is not tough to calculate.

1. Step 1. Find the present value of each annual coupon or principal payment [PV(Ct)]. Use the prevailing YTM on the bond as the discount rate.

2. Step 2. Divide this present value by the current market price of the bond (BP). This is the weight, or the fraction of the bond’s total value accounted for by each individual payment. Because a bond’s value is just the sum of the present values of its cash payments, these weights must sum to 1.0.

3. Step 3. Multiply this weight by the year in which the cash flow is to be received (t).

4. Step 4. Repeat steps 1 through 3 for each year in the life of the bond, and then add up the values computed in step 3.

Table 11.1 Duration Calculation for a 7.5%, 15-Year Bond Priced to Yield 8%

(1)	(2)	(3)	(4)	(5)
Year t	Annual Cash Flow Ct	Present Value at 8% of Annual Cash Flow (2)÷(1.08)t(2)÷(1.08)t	Present Value of Annual Cash Flow Divided by Price of the Bond (3)÷$957.20(3)÷$957.20	Time-Weighted Relative Cash Flow (1)×(4)(1)×(4)
1	$ 75	$ 69.44	0.0725	0.0725
2	$ 75	$ 64.30	0.0672	0.1344
3	$ 75	$ 59.54	0.0622	0.1866
4	$ 75	$ 55.13	0.0576	0.2304
5	$ 75	$ 51.04	0.0533	0.2666
6	$ 75	$ 47.26	0.0494	0.2963
7	$ 75	$ 43.76	0.0457	0.3200
8	$ 75	$ 40.52	0.0423	0.3387
9	$ 75	$ 37.52	0.0392	0.3528
10	$ 75	$ 34.74	0.0363	0.3629
11	$ 75	$ 32.17	0.0336	0.3696
12	$ 75	$ 29.78	0.0311	0.3734
13	$ 75	$ 27.58	0.0288	0.3745
14	$ 75	$ 25.53	0.0267	0.3735
15	$1,075	$338.88	0.3540	5.3106
		Price of Bond: $957.20	1.00	Duration: 9.36 yr

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Duration for a Single Bond

Table 11.1  illustrates the four-step procedure for calculating the duration of a 7.5%, 15-year bond priced at $957.20 to yield 8%.  Table 11.1  provides the basic input data: Column (1) shows the year t in which each cash flow arrives. Column (2) provides the dollar amount of each annual cash flow (Ct) (coupons and principal) made by the bond. Column (3) lists the present value of each annual cash flow in year t at an 8% discount rate (which is equal to the prevailing YTM on the bond). For example, in row 1 of  Table 11.1 , we see that in year 1 the bond makes a $75 coupon payment, and discounting that to the present at 8% reveals that the first coupon payment has a present value of $69.44. If we sum the present value of the annual cash flows in column (3), we find that the current market price of the bond is $957.20.

Investor Facts

Different Bonds, Same Durations Sometimes, you really can’t judge a book—or a bond, for that matter—by its cover. Here are three bonds that, on the surface, appear to be totally different:

· An 8-year, zero-coupon bond priced to yield 6%

· A 12-year, 8.5% bond that trades at a yield of 8%

· An 18-year, 10.5% bond priced to yield 13%

Although these bonds have different coupons and different maturities, they have one thing in common: they all have identical durations of eight years. Thus, if interest rates went up or down by 50 to 100 basis points, the market prices of these bonds would all behave pretty much the same!

Next, in column 4 we divide the present value in column 3 by the current market price of the bond. If the present value of this bond’s first coupon payment is $69.45 and the total price of the bond is $957.20, then that first payment accounts for 7.25% of the bond’s total value (i.e., $69.45÷$957.20 = 0.0725)(i.e., $69.45÷$957.20 = 0.0725) Therefore, 7.25% is the “weight” given to the cash payment made in year 1. If you sum the weights in column 4, you will see that they add to 1.0. Multiplying the weights from column 4 by the year t in which the cash flow arrives results in a time-weighted value for each of the annual cash flow streams shown in column 5. Adding up all the values in column 5 yields the duration of the bond. As you can see, the duration of this bond is a lot less than its maturity. In addition, keep in mind that the duration on any bond will change over time as YTM and term to maturity change. For example, the duration on this 7.5%, 15-year bond will fall as the bond nears maturity and/or as the market yield (YTM) on the bond increases.

Duration for a Portfolio of Bonds

The concept of duration is not confined to individual bonds only. It can also be applied to whole portfolios of fixed-income securities. The duration of an entire portfolio is fairly easy to calculate. All we need are the durations of the individual securities in the portfolio and their weights (i.e., the proportion that each security contributes to the overall value of the portfolio). Given this, the duration of a portfolio is the weighted average of the durations of the individual securities in the portfolio. Actually, this weighted-average approach provides only an approximate measure of duration. But it is a reasonably close approximation and, as such, is widely used in practice—so we’ll use it, too.

To see how to measure duration using this approach, consider the following five-bond portfolio:

Bond	Amount Invested *	Weight	×	Bond Duration	=	Portfolio Duration
*  Amount invested = Current market price × Par value of the bonds. That is, if the government bonds are quoted at 90 and the investor holds $300,000 in these bonds, then 0.90 × $300,000 = $270,000.
Government bonds	$ 270,000	0.15		6.25		0.9375
Aaa corporates	$ 180,000	0.10		8.90		0.8900
Aa utilities	$ 450,000	0.25		10.61		2.6525
Agency issues	$ 360,000	0.20		11.03		2.2060
Baa industrials	$ 540,000	0.30		12.55		3.7650
	$1,800.000	1.00				10.4510

In this case, the $1.8 million bond portfolio has an average duration of approximately 10.5 years.

If you want to change the duration of the portfolio, you can do so by (1) changing the asset mix of the portfolio (shift the weight of the portfolio to longer- or shorter-duration bonds, as desired) and/or (2) adding new bonds to the portfolio with the desired duration characteristics. As we will see below, this approach is often used in a bond portfolio strategy known as bond immunization.

Bond Duration and Price Volatility

A bond’s price volatility is, in part, a function of its term to maturity and, in part, a function of its coupon. Unfortunately, there is no exact relationship between bond maturities and bond price volatilities with respect to interest rate changes. There is, however, a fairly close relationship between bond duration and price volatility—as long as the market doesn’t experience wide swings in interest rates. A bond’s duration can be used as a viable predictor of its price volatility only as long as the yield swings are relatively small (no more than 50 to 100 basis points or so). That’s because as interest rates change, bond prices change in a nonlinear (convex) fashion. For example, when interest rates fall, bond prices rise at an increasing rate. When interest rates rise, bond prices fall at a decreasing rate. The duration measure essentially predicts that as interest rates change, bond prices will move in the opposite direction in a linear fashion. This means that when interest rates fall, bond prices will rise a bit faster than the duration measure would predict, and when interest rates rise, bond prices will fall at a slightly slower rate than the duration measure would predict. The bottom line is that the duration measure helps investors understand how bond prices will respond to changes in market rates, as long as those changes are not too large.

The mathematical link between changes in interest rates and changes in bond prices involves the concept of modified duration. To find modified duration, we simply take the (Macaulay) duration for a bond (as found from  Equation 11.8 ) and divide it by the bond’s yield to maturity.

Modified duration=(Macaulay) Duration in years1+Yield to maturityModified duration = (Macaulay) Duration in years1+Yield to maturityEquation11.9

Thus, the modified duration for the 15-year bond discussed above is

Modified duration=9.361+0.08=8.67––––––––––Modified duration = 9.361+0.08=8.67__

Note that here we use the bond’s computed (Macaulay) duration of 9.36 years and the same YTM we used to compute duration in  Equation 11.8 ; in this case, the bond was priced to yield 8%, so we use a yield to maturity of 8%.

To determine, in percentage terms, how much the price of this bond would change as market interest rates increased by 50 basis points from 8% to 8.5%, we multiply the modified duration value calculated above first by − (because of the inverse relationship between bond prices and interest rates) and then by the change in market interest rates. That is,

Percent changein bond price=−1×Modified duration×Change in interest rates=−1×8.67×0.5% =−4.33––––––––––Percent change in bond price=−1×Modified duration×Change in interest rates= −1×8.67×0.5% =−4.33__Equation11.10

Thus, a 50-basis-point (or ½ of 1%) increase in market interest rates will lead to an approximate 4.33% drop in the price of this 15-year bond. Such information is useful to bond investors seeking—or trying to avoid—price volatility.

Effective Duration

One problem with the duration measures that we’ve studied so far is that they do not always work well for bonds that may be called or converted before they mature. That is, the duration measures we’ve been using assume that the bond’s future cash flows are paid as originally scheduled through maturity, but that may not be the case with callable or convertible bonds. An alternative duration measure that is used for these types of bonds is the effective duration. To calculate effective duration (ED), you use  Equation 11.11 :

ED=BP(ri⏐↓)−BP(ri↑⏐)2×BP×ΔriED = BP(ri↓)−BP(ri↑)2×BP×ΔriEquation11.11

where

· BP(ri↑) = the new price of the bond if market interest rates go up

· BP(ri↓) = the new price of the bond if market interest rates go down

· BP = the original price of the bond

· Δri = the change in market interest rates

Example

Suppose you want to know the effective duration of a 25-year bond that pays a 6% coupon semiannually. The bond is currently priced at $882.72 for a yield of 7%. Now suppose the bond’s yield goes up by 0.5% to 7.5%. At that yield the new price would be $831.74 (using a calculator, N = 50, I = 3.75, PMT = 30, and PV = 1,000). What if the yield drops by 0.5% to 6.5%? In that case, the price rises to $938.62 (N = 50, I = 3.25, PMT = 30, PV = 1,000). Now we can use  Equation 11.11  to calculate the bond’s effective duration.

Effective duration = ($938.62−$831.74)÷(2×$882.72×0.005) = 12.11Effective duration = ($938.62−$831.74)÷(2×$882.72×0.005) = 12.11

This means that if interest rates rise or fall by a full percentage point, the price of the bond would fall or rise by approximately 12.11%. Note that you can use effective duration in place of modified duration in  Equation 11.10  to find the percent change in the price of a bond when interest rates move by more or less than 1.0%. When calculating the effective duration of a callable bond, one modification may be necessary. If the calculated price of the bond when interest rates fall is greater than the bond’s call price, then use the call price in the equation rather than BP(ri↓) and proceed as before.

Uses of Bond Duration Measures

You can use duration analysis in many ways to guide your decisions about investing in bonds. For example, as we saw earlier, you can use modified duration or effective duration to measure the potential price volatility of a particular issue. Another equally important use of duration is in the structuring of bond portfolios. That is, if you thought that interest rates were about to increase, you could reduce the overall duration of the portfolio by selling higher-duration bonds and buying shorter-duration bonds. Such a strategy could prove useful because shorter-duration bonds do not decline in value to the same degree as longer-duration bonds. On the other hand, if you felt that interest rates were about to decline, the opposite strategy would be appropriate.

Active, short-term investors frequently use duration analysis in their day-to-day operations. Longer-term investors also employ it in planning their investment decisions. Indeed, a strategy known as bond portfolio immunization represents one of the most important uses of duration.

Bond Immunization

Some investors hold portfolios of bonds not for the purpose of “beating the market,” but rather to accumulate a specified level of wealth by the end of a given investment horizon. For these investors, bond portfolio  immunization  often proves to be of great value. Immunization allows you to derive a specified rate of return from bond investments over a given investment interval regardless of what happens to market interest rates over the course of the holding period. In essence, you are able to “immunize” your portfolio from the effects of changes in market interest rates over a given investment horizon.

To understand how and why bond portfolio immunization is possible, you will recall from our earlier discussion that changes in market interest rates will lead to two distinct and opposite changes in bond valuation. The first effect is known as the price effect, and the second is known as the reinvestment effect. Whereas an increase in rates has a negative effect on a bond’s price, it has a positive effect on the reinvestment of coupons. Therefore, when interest rate changes do occur, the price and reinvestment effects work against each other from the standpoint of the investor’s wealth.

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Table 11.2 Bond Immunization

Year t	Cash Flow from Bond						Terminal Value of Reinvested Cash Flow
1	$ 80	×	(1.08)4	×	(1.06)3	=	$ 129.63
2	$ 80	×	(1.08)3	×	(1.06)3	=	$ 120.03
3	$ 80	×	(1.08)2	×	(1.06)3	=	$  111.14
4	$ 80	×	(1.08)	×	(1.06)3	=	$  102.90
5	$ 80	×	(1.06)3			=	$ 95.28
6	$ 80	×	(1.06)2			=	$ 89.89
7	$ 80	×	(1.06)			=	$ 84.80
8	$ 80					=	$ 80.00
8	$1,036.67						$1,036.67
			Total	$1,850.33
			Investor’s required wealth at 8%	$1,850.93
			Difference	$ 0.60

When the average duration of the portfolio just equals the investment horizon, these counteracting effects offset each other and leave your position unchanged. This should not come as much of a surprise because such a property is already embedded in the duration measure. If that relationship applies to a single bond, it should also apply to the weighted-average duration of a whole bond portfolio. When such a condition (of offsetting price and reinvestment effects) exists, a bond portfolio is immunized. More specifically, your wealth is immunized from the effects of interest rate changes when the weighted-average duration of the bond portfolio exactly equals your desired investment horizon.  Table 11.2  provides an example of bond immunization using a 10-year, 8% coupon bond with a duration of 8 years. Here, we assume that your desired investment horizon is also 8 years.

The example in  Table 11.2  assumes that you originally purchased the 8% coupon bond at par. It further assumes that market interest rates for bonds of this quality drop from 8% to 6% at the end of the fifth year. Because you had an investment horizon of exactly 8 years and desire to lock in an interest rate return of exactly 8%, it follows that you expect to accumulate cash totaling $1,850.93 [i.e., $1,000 invested at 8% for 8 years = $1,000×(1.08)8 = $1,850.93]8 years = $1,000×(1.08)8 = $1,850.93], regardless of interest rate changes in the interim. As you can see from the results in  Table 11.2 , the immunization strategy netted you a total of $1,850.33—just 60 cents short of your desired goal. Note that in this case, although reinvestment opportunities declined in years 5, 6, and 7 (when market interest rates dropped to 6%), that same lower rate led to a higher market price for the bond. That higher price, in turn, provided enough capital gains to offset the loss in reinvested income. This remarkable result clearly demonstrates the power of bond immunization and the versatility of bond duration. And note that even though the table uses a single bond for purposes of illustration, the same results can be obtained from a bond portfolio that is maintained at the proper weighted-average duration.

Maintaining a fully immunized portfolio (of more than one bond) requires continual portfolio rebalancing. Indeed, every time interest rates change, the duration of a portfolio changes. Because effective immunization requires that the portfolio have a duration value equal in length to the remaining investment horizon, the composition of the portfolio must be rebalanced each time interest rates change. Further, even in the absence of interest rate changes, a bond’s duration declines more slowly than its term to maturity. This, of course, means that the mere passage of time will dictate changes in portfolio composition. Such changes will ensure that the duration of the portfolio continues to match the remaining time in the investment horizon. In summary, portfolio immunization strategies can be extremely effective, but immunization is not a passive strategy and is not without potential problems, the most notable of which are associated with portfolio rebalancing.

Concepts in Review

Answers available at  http://www.pearsonhighered.com/smart

1. 11.10 What does the term duration mean to bond investors and how does the duration of a bond differ from its maturity? What is modified duration, and how is it used? What is effective duration, and how does it differ from modified duration?

2. 11.11 Describe the process of bond portfolio immunization, and explain why an investor would want to immunize a portfolio. Would you consider portfolio immunization a passive investment strategy comparable to, say, a buy-and-hold approach? Explain.

Bond Investment Strategies

1. LG 6

Generally speaking, bond investors tend to follow one of three kinds of investment programs. First, there are those who live off the income. They are conservative, quality-conscious, income-oriented investors who seek to maximize current income. Second, there are the speculators (bond traders). Their investment objective is to maximize capital gains, often within a short time span. Finally, there are the long-term investors. Their objective is to maximize total return—from both current income and capital gains—over fairly long holding periods.

In order to achieve the objectives of any of these programs, you need to adopt a strategy that is compatible with your goals. Professional money managers use a variety of techniques to manage the multimillion- (or multibillion-) dollar bond portfolios under their direction. These range from passive approaches, to semiactive strategies, to active, fully managed strategies using interest rate forecasting and yield spread analysis. Most of these strategies are fairly complex and require substantial computer support. Even so, we can look briefly at some of the more basic strategies to gain an appreciation of the different ways in which you can use fixed-income securities to reach different investment objectives.

Passive Strategies

The bond immunization strategies we discussed earlier are considered to be primarily passive in nature. Investors using these tools typically are not attempting to beat the market but to lock in specified rates of return that they deem acceptable, given the risks involved. As a rule, passive investment strategies are characterized by a lack of input regarding investor expectations of changes in interest rates and/or bond prices. Further, these strategies typically do not generate significant transaction costs. A buy-and-hold strategy is perhaps the most passive of all investment strategies. All that is required is that the investor replace bonds that have deteriorating credit ratings, have matured, or have been called. Although buy-and-hold investors restrict their ability to earn above-average returns, they also minimize the losses that transaction costs represent.

One popular approach that is a bit more active than buy-and-hold is the use of  bond ladders . In this strategy, equal amounts are invested in a series of bonds with staggered maturities. Here’s how a bond ladder works. Suppose you want to confine your investing to fixed-income securities with maturities of 10 years or less. Given that maturity constraint, you could set up a ladder by investing (roughly) equal amounts in, say, 3-, 5-, 7-, and 10-year issues. When the 3-year issue matures, you would put the money from it (along with any new capital) into a new 10-year note. You would continue this rolling-over process so that eventually you would hold a full ladder of staggered 10-year notes. By rolling into new 10-year issues every 2 or 3 years, the interest income on your portfolio will be an average of the rates available over time. The laddered approach is a safe, simple, and almost automatic way of investing for the long haul. A key ingredient of this or any other passive strategy is, of course, the use of high-quality investments that possess attractive features, maturities, and yields.

Trading on Forecasted Interest Rate Behavior

In contrast to passive strategies, a more risky approach to bond investing is the forecasted interest rate approach. Here, investors seek attractive capital gains when they expect interest rates to decline and preservation of capital when they anticipate an increase in interest rates. This strategy is risky because it relies on the imperfect forecast of future interest rates. The idea is to increase the return on a bond portfolio by making strategic moves in anticipation of interest rate changes. Such a strategy is essentially market timing. An unusual feature of this tactic is that most of the trading is done with investment-grade securities because these securities are the most sensitive to interest rate movements, and that sensitivity is what active traders hope to profit from.

This strategy brings together interest rate forecasts and the concept of duration. For example, when a decline in rates is anticipated, aggressive bond investors often seek to lengthen the duration of their bonds (or bond portfolios) because bonds with longer durations (e.g., long-term bonds) rise more in price than do bonds with shorter durations. At the same time, investors look for low-coupon and/or moderately discounted bonds because these bonds have higher durations, and their prices will rise more when interest rates fall. Interest rate swings may be short-lived, so bond traders try to earn as much as possible in as short a time as possible. When rates start to level off and move up, these investors begin to shift their money out of long, discounted bonds and into high-yielding issues with short maturities. In other words, they do a complete reversal and look for bonds with shorter durations. During those periods when bond prices are dropping, investors are more concerned about preservation of capital, so they take steps to protect their money from capital losses. Thus, they tend to use such short-term obligations as Treasury bills, money funds, short-term (two- to five-year) notes, or even variable-rate notes.

Bond Swaps

In a  bond swap , an investor simultaneously liquidates one position and buys a different issue to take its place. Swaps can be executed to increase current yield or yield to maturity, to take advantage of shifts in interest rates, to improve the quality of a portfolio, or for tax purposes. Although some swaps are highly sophisticated, most are fairly simple transactions. They go by a variety of colorful names, such as “profit takeout,” “substitution swap,” and “tax swap,” but they are all used for one basic reason: portfolio improvement. We will briefly review two types of bond swaps that are fairly simple and hold considerable appeal: the yield pickup swap and the tax swap.

In a  yield pickup swap , an investor switches out of a low-coupon bond into a comparable higher-coupon issue in order to realize an instantaneous pickup of current yield and yield to maturity. For example, you would be executing a yield pickup swap if you sold 20-year, A-rated, 6.5% bonds (which were yielding 8% at the time) and replaced them with an equal amount of 20-year, A-rated, 7% bonds that were priced to yield 8.5%. By executing the swap, you would improve your current yield (your interest income would increase from $65 a year to $70 a year) as well as your yield to maturity (from 8% to 8.5%). Such swap opportunities arise because of the yield spreads that normally exist between different types of bonds. You can execute such swaps simply by watching for swap candidates and asking your broker to do so. In fact, the only thing you must be careful of is that transaction costs do not eat up all the profits.

Another popular type of swap is the  tax swap , which is also relatively simple and involves few risks. You can use this technique whenever you have a substantial tax liability as a result of selling some security holdings at a profit. The objective is to execute a swap to eliminate or substantially reduce the tax liability accompanying the capital gains. This is done by selling an issue that has undergone a capital loss and replacing it with a comparable obligation.

For example, assume that you had $10,000 worth of corporate bonds that you sold (in the current year) for $15,000, resulting in a capital gain of $5,000. You can eliminate the tax liability accompanying the capital gain by selling securities that have capital losses of $5,000. Let’s assume you find you hold a 20-year, 4.75% municipal bond that has undergone a $5,000 drop in value. Thus, you have the required tax shield in your portfolio. Now you need to find a viable swap candidate. Suppose you find a comparable 20-year, 5% municipal issue currently trading at about the same price as the issue being sold. By selling the 4.75s and simultaneously buying a comparable amount of the 5s, you will not only increase your tax-free yields (from 4.75% to 5%) but will also eliminate the capital gains tax liability.

The only precaution in doing tax swaps is that you cannot use identical issues in the swap transactions. The IRS would consider that a “wash sale” and disallow the loss. Moreover, the capital loss must occur in the same taxable year as the capital gain. Typically, at year-end, tax loss sales and tax swaps multiply as knowledgeable investors hurry to establish capital losses.

Concepts in Review

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1. 11.12 Briefly describe a bond ladder and note how and why an investor would use this investment strategy. What is a tax swap and why would it be used?

2. 11.13 What strategy would you expect an aggressive bond investor (someone who’s looking for capital gains) to employ?

3. 11.14 Why is interest sensitivity important to bond speculators? Does the need for interest sensitivity explain why active bond traders tend to use high-grade issues? Explain.