Inflation is generally described as an increase in the prices of goods and services generally, but that is a misleading way of thinking about it. Inflation is more appropriately thought of as a decline in one price – the purchasing power of the unit of account, the dollar.

To see how it works, let's do a thought experiment. In the case of distance measurement, the most widely used unit of account is the meter. Until the twentieth century, the meter was defined to be the length of a platinum bar kept in an air-tight, temperature-controlled vault in Paris. If a dispute arose regarding whether some object was a meter long, in principle the vault could be opened and the bar could be used to confirm whether the object in question was the same length.

Now assume that some alien force is able to start shrinking the platinum bar as part of an effort to confuse humanity. How would we experience the shrinkage? By definition, the bar in the vault would always be a “meter.” But when the official meter is used to measure the length of other objects they would all appear to be getting longer. Suppose, for instance, that the bar was shrinking at 2% per year. In that case, every object would be measured as being 2% longer each year. This could get confusing because true lengths could also be changing for other reasons. For example, a growing child would be measured as taller both because the child was growing and because the meter was shrinking.

One way to confront the complexity would be to define what might be called a “constant year meter.” For instance, a constant year meter could be defined as the length of the bar on January 1, 2017. Other lengths could then be stated in terms of “2017 meters.” That would be a way to get around the shrinkage problem.

Now it may seem to you like this is all a bunch of theoretical nonsense. And from the point of view of the meter, that is probably right. But with respect to the dollar, it is exactly what has been happening almost every year for the past century. As the dollar shrinks, we experience it as increases in the price of goods and services because all of those prices are measured in dollars. In addition, like the growing child, prices change both because the dollar is shrinking and because the prices are moving for other reasons. Therefore, the Bureau of Labor Statistics (BLS), which keeps track of consumer goods and services prices, does, in fact, use a constant year dollar. The BLS speaks of prices defined in terms of both current dollars (today's unit) and constant year dollars (reflecting the purchasing power of the dollar in 2010).

To measure the change in prices generally, the BLS constructs what is known as the Consumer Price Index, or CPI. The CPI measures the cost of purchasing a diverse bundle of goods and services. The index is adjusted to take account of the fact that over time the make-up and quality of the goods and services that people consume changes. The inflation rate is defined to be the percentage change in the consumer price index.

The reason that taking account of inflation is so important to investors is because people consume goods and services, not dollars. Because the dollar is constantly shrinking, you can't measure the success of investment by simply looking at the path of wealth (POW) measure in terms of dollars. The dollars at the end of the period are worth less than the dollars at the beginning. Instead, it is necessary to work with what are called real, or inflation-adjusted, returns. A real POW can then be constructed using the real returns. By way of distinction, returns and POWs stated in terms of current dollars are referred to as nominal. The easiest way to understand how all this works is by going through an example. But before that, let's get started by looking at some actual data on inflation.

The first column of Exhibit 3.1 presents the annual rate of inflation from 1926 to 2017. The exhibit shows that inflation has been far from constant in the United States. At the beginning of the sample period, inflation was negative (there was deflation) for the first seven years. The rate of deflation reached 10% at the height of the great depression in 1932. With the onset of World War II, inflation turned sharply positive, rising as high as 18.13% in 1946. During the 1950s and early 1960s, inflation was quiescent, averaging less than 2%. Then it turned up again during the escalation of the Vietnam War. From 1973 until 1981, the United States experienced its most prolonged period of significant inflation when the average was close to 10%. In recent years, following the financial crisis, inflation has fallen back to less than 2% per year. Currently, the Federal Reserve has an inflation target of 2% per year. Over the full period from 1926 until 2017, the arithmetic average rate of inflation was 2.97%.

The second column of the exhibit shows the value of a dollar for each current year in terms of 1926 dollars. Notice that by 1933 the purchasing power of a current dollar had risen to $1.356 in terms of 1926 dollars. But with the onset of inflation the dollar began to shrink. By 2017, the purchasing power of a current dollar was worth only 7.3 cents in terms of 1926 dollars.

It is worth noting that although inflation in the United States ranged from about −10% to 10% that variation is small by international and historical standards. Inflation rates of 100% per year or more are not that rare and inflation exceeding 1,000,000% per year have been recorded in recent decades. As we write this, 100 trillion dollar notes in Zimbabwe dollars can be purchased on eBay for about $20 U.S. dollars. In Germany after the First World War, prices rose by a factor of 10¹², aiding Hitler's rise to power.

To illustrate how inflation impacts the real returns on investments, the second column of Exhibit 3.2 starts by reproducing the nominal returns on the Center for Research in Securities Prices (CRSP) market index from the chapter on returns. Given the nominal returns, the formula for computing the real return is

(3.1)

Applying this formula leads to the series of real returns on the market shown in column three. Finally, the POW is computed using both the nominal returns in column four and the real returns in column five. The results show that the impact of inflation is significant. In nominal terms $1 invested in 1926 rises to $5,599.04. But in terms of 1926 dollars, the 2017 value of the investment is only $395.65. That lower figure represents the actual increase in consumable wealth.

Exhibit 3.3 plots the POW for the CRSP market index in both nominal and real terms. Notice that it is when the periods of high inflation begin that the lines diverge most noticeably. The long-run impact of the compounding of inflation comes through clearly.

By tradition, most investment performance results are stated in terms of nominal returns. This can cause a good deal of mischief. High returns in a high-inflation environment may well provide less increase in consumable wealth than low returns with low inflation. As we shall see later, inflation also interacts with taxes because most taxes are levied in nominal terms. This also impacts consumable wealth. The bottom line is that inflation is not something investors can ignore, even when it is as low as 2%.

The relation between security returns and inflation is particularly important in the case of fixed income securities. Inflation impacts fixed income securities in two ways. First, the rate of inflation expected over the life of a fixed income security is a prime determinant of the rate of interest paid by the security. Second, the real purchasing power of the dollars received from the security depends on the subsequent rate of inflation that occurs over the life of the security. To explain this further, it is first necessary to know more about how fixed income securities work. There is no better place to start than with the most important fixed income securities in the world – U.S. Treasury bonds and U.S. Treasury bills.

TREASURY BILLS AND TREASURY BONDS

Because the United States has run large deficits for decades there are now over $20 trillion dollars of Treasury securities outstanding. Ironically, the debt is almost exactly equal to the total value of stocks traded on the New York Stock Exchange. These securities are generally referred to as risk free because the U.S. government stands behind them and could, if necessary, print money to redeem them. But with the exception of Treasury Inflation Protected Securities, or TIPS, they are risk free only in nominal terms. There is no certainty regarding the purchasing power of the dollars that investors will receive.

Exhibit 3.4 shows that there are two types of Treasury securities: Treasury bills and Treasury notes and bonds. Treasury bills have a maturity of one year or less, while Treasury notes range in maturity from 2 to 10 years and Treasury bonds have maturities from 10 to 30 years.

The data reported in the exhibit are based on prices quoted in the secondary market. After their initial auction, Treasury securities are actively traded through a network of government securities dealers. Each of these dealers quotes prices at which the firm is willing to buy or sell outstanding Treasury issues. The dollar volume of trading in Treasury securities outstrips the volume of trading on the New York Stock Exchange despite the fact that there are a relatively limited number of issues. This makes the market one of the most liquid in the world.

Treasury Bills

Treasury bills are discount securities, which means they are issued at a price less than par and appreciate to par at maturity. For example, a 91-day Treasury bill with a par value of $10,000 discounted at 2.00% will be issued at a price of $9,949.44. Between the issuance date and maturity no other payments are made to the holders of the bills. The Treasury issues bills with three different maturities: three months, six months, and one year. New three- and six-month bills are offered each week; one-year bills are offered once a month. Because the secondary market is highly liquid, bills with any maturity up to one year can easily be purchased with minimal transaction costs.

The bid and ask quotes for Treasury bills, such as those shown in the far-right-hand column of Exhibit 3.4, are stated in terms of the “bank discount” which is quoted on a 360-day basis. The bid price is the price at which a government security dealer is willing to purchase the security and the ask price is the price at which a government security dealer will sell the security. The bid and ask quotes reported in the Wall Street Journal are indicative. Actual bid-ask spreads on round lot trades of $10 million or more, which are obtained by contacting government securities dealers, typically are narrower.

The bank discount and the price of the bill per hundred dollars of par value are related by the formula:

(3.2)

where,

• p = the price of the Treasury bill per $100 par value,
• d = the bank discount, and
• n = the number of days until the bill matures.

For example, consider the bill maturing February 15, 2018. As of November 20, 2017, that bill had a maturity of 87 days and was selling at an ask bank discount of 1.245. Substituting these numbers into Eq. (3.2) gives:

Though it looks like an interest rate when reported in the newspaper, the bank discount is not a yield; it is simply a convention used by government securities dealers to quote prices. For this reason, bond equivalent yields on Treasury bills are also reported. These bond equivalent yields, which are shown in the last column of Exhibit 3.4, are comparable to the yield to maturity (YTM) on bonds. The bond equivalent yield for a bill with a maturity of 182 days or less is calculated using the formula:

(3.3)

where the variables are as defined above. Bond equivalent yields are calculated from the perspective of a buyer of securities and, therefore, are based on ask prices. Utilizing the data above for the 87-day bill, the bond equivalent yield comes to 1.266% as shown in Exhibit 3.4.

Neither the bank discount nor the bond equivalent yield measures the true annual return on a Treasury bill. The true annual return is that rate which an investor would earn over the course of a year if he or she invested in Treasury bills of a given type and reinvested the proceeds in identical bills until the end of the year. Because of the assumed reinvestment, the true annualized yield takes account of compound interest.

The true annualized yield on a Treasury bill can be computed by a two-step procedure. First, Eq. (3.2) is solved for the price of the bill. Second, the true annualized yield is calculated from the price (per hundred), p, the par value, 100, and the maturity in days, n, using the familiar compounding formula:

(3.4)

Applying Eq. (3.4) to the 85-day bill with a price of 99.119 discussed above, the true annualized yield is 1.27%. Notice that the true annualized yield is at most 2.5 basis points greater than the bank discount and less than 1 basis point greater than the bond equivalent yield. Though these differences are small, they can grow rapidly as rates rise. For example, at a bank discount on the order of 12.00 for 90-day bills, as existed during the late 1970s, the discrepancy between the discount and the true annualized yield is over 50 basis points. When the Treasury bill rate is used as the “risk-free” rate in an economic model, the appropriate measure of return is the true annualized yield because it takes proper account of the compounding of interest.

Treasury Notes and Bonds

Unlike Treasury bills, Treasury notes and bonds pay interest in the form of semi-annual coupons. The bonds come in units with a par value of $1000, but the prices are quoted per hundred. For example, consider the last Treasury bond listed in Exhibit 3.4, the 2.75% coupon maturing in November 2047. As of November 20, 2017, the date on which the data were collected by the Wall Street Journal, this was the most recently issued 30-year Treasury bond. The most recently issued 30-year bond is often called the “long” bond or the “bellwether” bond and is closely followed as a benchmark of the market for long-term funds. As shown in the table, the long bond is quoted at prices of 99.195 bid (per hundred) and 99.227 ask.

The semi-annual interest payments on a Treasury bond are determined by the coupon rate. Each interest payment equals the coupon rate times the par value of the bond divided by two. In the case of the long bond, the coupon is 2.75%, which means that the bond pays $13.75 in interest per $1000 in par value every six months for the next 30 years.

The final number shown in the right-hand column is the yield to maturity (YTM). The YTM is defined as twice the internal rate of return, calculated at the ask price. The factor of two is included to adjust for the fact the payments are received semi-annually. As will become apparent, however, this adjustment does not take account of compound interest.

The internal rate of return (IRR) on an investment is that rate which discounts the cash flows from the investment back to the purchase price.¹ It can be quickly computed using spreadsheet software such as Excel. For instance, the internal rate of return on a $100 investment which provides three annual payments of $40, $50 and $60 is the discount rate r, which solves the equation,

(3.5)

Application of the IRR function in Excel reveals that r equals 21.6%.

In the case of Treasury bonds, the IRR is that rate which discounts the promised stream of interest and principal payments back to the price of the bond. Returning, for instance, to the long bond and assuming that its maturity is exactly 30 years, the bond provides 60 interest payments of $13.75 and a final principal payment of $1,000 at maturity. The current ask price of the bond is 99.226 per hundred or $992.26 per bond. Thus the IRR is that rate which discounts a stream involving 59 payments of $13.75 and then one payment of $1,013.75 back to a present value of $99.26. Using Excel that rate is found to be 1.394%. Because the cash flows are semi-annual, the resulting internal rate of return is a semi-annual rate. This semi-annual rate is multiplied by two and rounded to two decimal places to calculate what is called the YTM on the bond, which is 2.788% in the example. Bond traders usually quote bond “prices” in terms of YTM. This is because the number is more useful for investors who are concerned with the rate they will earn if they buy the bond, not the price of the bond per hundred dollars of par value.

As noted above, the calculation of the YTM ignores compound interest. To calculate the true annualized yield on a bond, account must be taken of compounding.

The internal rate of return of 1.394% is a true yield, but it is stated per six months, not per year. Using the compounding formula, the annualized yield is given by

Applying the formula to a semi-annual rate of 1.394% gives a true annualized rate of 2.808%, which is 2 basis points greater than the YTM.

All the foregoing comments and calculations apply to high-grade corporate bonds as well as Treasury bonds. For now, high grade means that the probability of default is so small it can be ignored. Like Treasuries, high-grade corporate bonds typically pay coupons semi-annually and their YTM is calculated in the same way.

INTEREST RATES AND INFLATION

In our discussion of the relation between interest rates and inflation, we focus on Treasuries and other high-grade bonds. Because there is no risk of default, the prices of these securities are not affected by credit risk, which adds another layer of complexity. We turn to the impact of credit risk later in the chapter.

Because the payouts are fixed at the time they are issued, the real rate of return an investor earns on a bond or bill will depend upon the rate of inflation between the time an investor buys the security and the time it is redeemed. To illustrate, consider the case of a one-year Treasury bill issued at a discount of 2.000%. Using Equations (3.1) and (3.3) above, the annualized interest rate on this security is 2.070%. If the subsequently realized inflation rate is 1.75%, then from Eq. (3.1) the realized real return is 0.314%.

Because it is the real return that determines their consumable wealth, investors will take account of the inflation they expect at the time they purchase the Treasury bill. If inflation was expected to be 10%, it would make little sense to buy a bill yielding 2.5%. For this reason, interest rates on fixed income securities reflect expected inflation. Unfortunately, expected inflation, like expected returns, cannot be measured directly, but it is presumably related to recent past inflation. Therefore, in high-inflation environments there should be high interest rates and in low-inflation environments there should be low interest rates. This turns out to be the case.

Exhibit 3.5 provides an illustration. The bars show the rate of inflation for the year in question, the black line plots the interest on short-term Treasury bills over the years from 1960 to 2017. Notice how the interest rate tends to follow the rate of inflation. When inflation rises, inflationary expectations follow and so do interest rates. Although inflation is not the only thing that affects interest rates, academic studies indicate it is the most important. Exhibit 3.5 is consistent with that finding. Interest rates are high when inflation is high and low when inflation is low.

The relation between interest rates and expected inflation is given by an analog to Eq. (3.1). The relation states that

For interest rates and inflation rates less than 10%, such as those experienced in the United States, the equation can be approximated by

(3.6)

Equation (3.6) provides insight into why most of the variation in nominal interest rates (the financial media generally omit the word nominal and just say interest rates) is due to changes in inflation. Real interest rates are determined by fundamental economic forces and move within a relatively tight range. Inflation rates are determined by government policies, which can vary dramatically. As inflation rates vary, so does expected inflation and, from Eq. (3.6), the level of nominal interest rates. Equation (3.6) explains what we see in Exhibit 3.5. It is also a warning that if inflation were to spike up from the current low rates of less than 2.0%, nominal interest rates would follow.

The foregoing observations are not limited to interest rates on Treasury securities. The forces that set equilibrium in the financial markets are real forces related to real rates of return. When inflation rises, investors will buy securities only if nominal expected returns rise by a comparable amount so that the expected real returns remain largely unchanged.

The foregoing points to an error often made by the financial media. They compare nominal rates of return over time. But given the variation in the rate of inflation documented in Exhibit 3.5, nominal rates should not be constant over time. Periods with higher inflation should have higher nominal returns on all assets. This is particularly true internationally where inflation rates can vary immensely. It is for this reason, unless otherwise noted, that we will specify whether real or nominal returns are used in making investment comparisons. That is a good habit to adopt.

BOND YIELDS AND BOND RETURNS

Let us set inflation aside for a moment to delve into the difference between bond yields and bond returns. We spent the whole first chapter stressing the importance of returns and have yet to mention them in our discussion of bonds. It turns out that the fact that bonds provided a fixed sequence of payments has implications for bond returns. The best way to see how it all operates is by working through a detailed example.

Consider an investor who buys a 10-year bond with a 3% coupon rate at a price of $100 dollars per $100 dollars in maturity value. When a bond sells for its maturity value it is said to be trading at par. The investor who purchases the bond will receive twenty semi-annual payments of $1.50 and a final principal payment of $100 for every $100 invested. To calculate the YTM, we need a more general version of Eq. (3.5). That more general version is given by Eq. (3.7), which relates the bond price, coupon, and YTM.

(3.7)

In Eq. (3.7), P is the price of the bond, the Cs are the semi-annual coupons, Prin is the principal paid at maturity, n is the maturity of the bond measured as the number of semi-annual periods, and YTM is the yield to maturity. Because the coupons and the principal are defined by the bond contract when a bond is issued, there are only two unknowns in Eq. (3.7): the price and the YTM. Although the equation is too complex to solve algebraically, a spreadsheet will quickly find the solution. Plugging in the 20 coupons of $1.50, the principal value of $100, and the price of $100 gives a YTM of 3.00%. That is not a fluke. In general, when a bond is selling at par the YTM equals the coupon rate.

Although we started with the price and solved for the YTM, that is not the way the bond market typically operates. Instead the level of yields in market is set by factors such as the level of macroeconomic activity, Federal Reserve policy, and particularly the rate of inflation. The price of a particular bond is then set so that its yield is equal to the general level of yields.

To continue the example, suppose that five years after the investor bought the bond, when it still has five years left to maturity, interest rates in the economy jump. New five-year bonds selling at par now carry a coupon rate of 5%. To remain competitive with the new bonds, the old bond must also offer a YTM of 5%. How does it do that? By selling for a lower price. Holding the coupon constant, as the price of a bond falls, the YTM rises. For a five-year bond with a 3% coupon to have a yield of 5%, its price must be $91.25. At that price, the investor will earn 5% for the remaining five years. It may sound like that is too good to be true. The investor earns 3% for the first five years and 5% for the last five years on a 3% coupon bond purchased at par. The mistake is that the return for the first five years was not 3% because the investor experienced a capital loss. The investment of $100 is now worth only $91.25. Taking account of the capital loss the return over the first five years is actually only 1.30%.² As a result, the total return on the bond over the full 10 years is 3%, as it must be.

Exhibit 3.6 presents both the nominal and real returns on short-term Treasury bills and 20-year Treasury bonds over the period from 1960 to 2017. Notice that the Treasury bill nominal returns are never negative. Because the bills mature in a month, there is no opportunity for capital loss. The annual returns on the 20-year Treasury bonds are much more variable. This is because after one year a 20-year bond still has 19 years left until maturity. As a result, its price, and hence the investor return, depends on what has happened to yields on 19-year bonds in the past year. If market yields rise sharply, the bond price has to fall as it did in the example calculation. If the price drop is big enough, the annual return is negative. However, over the long run, coupon rates on new bonds adjust to reflect the new, higher interest rates. As a result, the average returns on 20-year Treasury bonds are higher, both in real and nominal terms, than those on Treasury bills, as they should be to reflect the risk associated with the greater price variation in the longer-term bonds.

TREASURY INFLATION PROTECTED SECURITIES (TIPS)

In addition to its nominal bonds, the U.S. Treasury also sells inflation-protected securities, or TIPS. These securities pay a fixed real rate of interest plus an added payment that compensates the investor for the realized rate of inflation. Consequently, the real rate that investors will earn on a TIPS is known, but the nominal rate is not because it depends on the subsequently realized inflation.

Because 10-year TIPS are relatively actively traded, comparing the yield on 10-year TIPS with the yield on 10-year nominal U.S. Treasury bonds produces a measure of the bond market's expectation of inflation over the next 10 years. To illustrate, Exhibit 3.7 plots the yield on 10-year TIPS and 10-year nominal bonds over the period from 2004 to 2017. The first thing that jumps out from the exhibit is that in the years 2012 and 2013, the yield on the TIPS (the 10-year real interest rate) was negative. Investors who bought these bonds and held them to maturity were assured of experiencing a decrease in purchasing power. This does not mean that the nominal yield was negative, because the investors received an added payment based on the rate of inflation. Nonetheless, it is surprising that investors were willing to accept negative real rates.

The second thing to note is that for most of the period the gap between the nominal yield and the TIPS yield is around 2% – a bit more at the beginning and a bit less at the end. This implies that investors were generally expecting future inflation to be about 2% over the 10 years following the date on which the gap was observed. Such expectations are consistent with the forecasts of most major economic forecasting firms and with the Federal Reserve's inflation target. The exception to the rule is the period at the heart of the financial crisis in 2009, when the expected inflation fell to zero. This suggests that investors were fearing a prolonged recession that would eliminate inflation. When those fears diminished, inflation expectations returned to close to 2%. Finally, the exhibit clearly shows how nominal interest rates track expected inflation and that most of the variation in nominal rates is due to changes in expected inflation. This can be seen by noting that the gap between the two lines, which measures the expected rate of inflation, varies by a lot less than either of the individual lines.

Looking back at Exhibit 3.6, you can see that with an expected inflation rate of around 2%, the expected real returns on Treasury bills were negative throughout the period following the financial crisis. This pattern of expected negative real rates on short-term financial instruments has been due in large part to the aggressive financial policies of world central banks, including the Federal Reserve. It is anomalous by historical standards. For example, Exhibit 3.6 shows that prior to the financial crisis the average real return on Treasury bills over the period from 1960 to 2007 was 1.30%, whereas following the crisis in the years 2008 to 2017 the average real return was −1.32%.

These extraordinarily low real interest rates had two effects worth noting. One is that they made life difficult for savers. For example, living off retirement savings becomes a lot more difficult when the real return on fixed income securities is negative. This has led many savers to “reach for yield” by buying increasingly risky securities. Because the economy has remained solid since the end of the crisis, this strategy has not caused problems. But if another recession occurs, it could turn into a disaster for many savers. Second, the big beneficiary of the low rates has been the world's biggest debtor – the United States government. But there are storm clouds here as well. If interest rates return to more normal levels, it will become a good deal more difficult for the government to pay the interest on its massive and growing debt.

A look back at Exhibit 3.6 seems to contradict part of the story regarding low interest rates because the average real return on bonds in the years following the crisis is 6.99%, well above its long-run average of 2.14%. The answer to the apparent paradox is again changes in the prices of the long-term bonds. As interest rates declined, the capital gains on the bonds associated with the drop produced rising prices for long-term bonds. Those rising prices produced the high annual returns. That, however, is a one-time event. Once the prices of bonds adjust, future returns will be in line with the current nominal yields, which are about 2.5% for 20-year Treasury bonds.

CORPORATE BONDS AND CREDIT RISK

To this point, it has been assumed that all the promised payments made on a fixed income security will be paid. This is a reasonable assumption for Treasury securities because the government has the power to raise taxes and print money. It is also a reasonable assumption for highly successful companies like Apple, whose operating earnings far exceed the interest on the debt. However, it is not reasonable for many smaller and more financially stressed companies. For bonds issued by such companies, it is necessary to draw a distinction between the payments the company promises to make and the payments the investor actually expects to receive. This leads to a corresponding difference between the promised yield and the expected yield. Understanding the distinction is critical for investors considering holding riskier corporate bonds.

Like Treasury bonds, most corporate bonds are defined by four parameters: the par value, the coupon rate, the maturity, and the price. The first three parameters are set by the issuing company and are stated in the bond indenture. The final parameter is determined by the market. The price adjusts until the yield on the bond is competitive with other bonds of similar risk available in the market.

Virtually all corporate bonds follow several standard conventions. First, the par value to be repaid at maturity is $1,000. Second, coupons are paid semi-annually. As for Treasury bonds, the coupon payments equal the coupon rate times the par value divided by two. Third, bond prices are stated per hundred dollars of par value. Finally, dealers typically quote the bonds in terms of YTM.

PROMISED YIELDS VERSUS EXPECTED YIELDS

The yields to maturity we have been calculating so far are better described as promised yields. The YTM is the yield the investor will earn over the life of a bond if all the payments are made as promised. But as we expand the scope to consider borrowers who may not honor their commitment, a gap begins to open between the promised yield and the yield an investor actually expects to earn. This gap reflects the credit risk of the issuer. The less likely it is that the issuer will honor its commitments, the larger the gap.

If a company finds itself unable to honor its commitments as they come due, the normal solution is a restructuring of the company's financing either voluntarily or in the context of a formal bankruptcy. In a restructuring, it is generally the case the bondholders have to accept less than their promised payments. Given that investors recognize that borrowers who represent a credit risk may have to restructure, they will develop estimates of their expected yield that depend on the probability of a restructuring and the amount that they expect to recover if restructuring becomes necessary. The expected yield will be less than the promised yield because investors will take account of the possibility that interest and principal payments may be delayed or omitted.

The promised yield and the expected yield are both ex-ante measures, meaning that they are calculated before the bond matures or defaults. The actual return the investor earns over the course of an investment cannot be calculated until the bond has matured or restructured, so that all the cash payments are known. At that time, the actual return can be computed as the internal rate of return which discounts the stream of payments received to the purchase price of the bond. The actual return calculated in this fashion need not equal either the promised yield or the expected yield. For instance, a bond may have a promised yield of 8%, and an expected yield of 5%, but the actual yield may be only 1% because of an early default.

Bonds with credit risk can be thought of as being a hybrid between bonds without credit risk and equity. Like bonds without credit risk, the cash payouts on bonds with credit risk are limited by the fixed payments specified in the bond contract. They can never exceed the coupons and the principal. However, like equity, the actual yield earned on a bond with credit risk depends on the performance of the company. If the company does well and all the payments are made as promised, the investor reaps an unexpected windfall. On the other hand, if the company performs poorly and a restructuring is required, the investor will typically earn even less than the ex-ante expected yield.

Because of the importance of credit risk assessment to bond investors, an industry has grown up to provide bond ratings. Bond rating firms collect detailed financial data on issuers and analyze the information in order to sort bonds into ratings buckets. The two leading bond rating firms are Standard and Poor's and Moody's. The rating classifications used by Standard and Poor's run: AAA, AA, A, BBB, BB, B, CCC, CC, C, and D. The classifications used by Moody's has the same number of buckets but they are labeled slightly differently, as follows: Aaa, Aa, A, Baa, Ba, B, Caa, Ca, C, D.

By convention, bonds with a rating of BBB (or Baa) and above are considered to be investment grade. This cut-off is important because there are regulations that require some institutional bondholders to limit their holdings to investment grade bonds. Bonds with a rating of BB or below are often referred to as “junk bonds” because of the risk of restructuring. Investment firms that deal in the bonds are not fond of that name. They prefer the label “high yield bonds” because the bonds have high promised yields. The two names, however, refer to the same thing. To avoid confusion, we will generally use the phrase “low grade” in the remainder of the book to refer to bonds with a rating below BBB.

As you might expect, as a bond's rating declines its promised yield rises to account for the fact that the ratings agencies believe that a restructuring is more likely. As evidence of this relation Exhibit 3.8 shows the promised yields on bonds by rating classification as of December 1, 2017.The data in the exhibit are from Merrill Lynch and refer to ratings by Standard & Poor's. As expected, the promised yields rise as the ratings fall from a low of 2.94% for AAA bonds, to 3.58% for BBB, to 5.83% for B, and all the way up to 10.76% for CCC. It bears repeating that the actual earned yield spreads will be less than the promised spreads. The CCC promised yields are considerably higher than those for AAA bonds because the market believes the probability of restructuring is significant for such low-rated bonds.

Exhibit 3.8 shows the promised yields for bonds with different ratings at one point in time. However, spreads between the yields on high-grade bonds and low-grade bonds are far from constant over time. Exhibit 3.9 plots the spread between the yield on 10-year Treasury bonds and the promised yield on a diversified index of low-grade bonds maintained by Merrill Lynch. The index includes bonds with ratings from BB to CCC. What is most striking about the plot is how much the spread varies. During the height of the financial crisis the spread exceeded 20%. That means that lower-grade-bond issuers had to promise to pay yields approaching 25%! Clearly, investors did not expect to earn a 25% return. They demanded such a high promised yield because they felt there was a substantial probability of default. By 2017, the spread had dropped all the way to below 4%. Such a low spread implies that in 2017 investors believed the probability of default was a fraction of what it was during the crisis years.

Whereas promised yields and spreads can be observed directly, expected returns and expected spreads are unobservable. What investors expect to earn on a given bond depends on how likely they think restructuring is, how soon they believe a restructuring might occur, and what they can expect to receive in a restructuring. None of these are observable. However, by looking at history, we can get an estimate of the market's expected return spread compared to Treasuries. If investor expectations are fulfilled on average, then the historical average return spread will approximate the expected return spread. In the case of the Merrill Lynch low-grade-bond index, over the last 10 years the average return has been about 3% above the average return on Treasury bonds compared to an average promised yield spread of more than 5%. This means that losses due to restructurings averaged about 2% per year.

Exhibit 3.10 presents additional data, collected by Professor Edward Altman, on promised yield spreads compared to actual return spreads. Although the data are from an earlier period, they are still useful in illustrating the concepts. The data show high variation in the promised spread similar to that observed in more recent years. The return data are even more variable because of the impact of capital gains and losses. The exhibit shows that the average yield spread was 4.55%, whereas the average return spread was only 2.88%. The numbers are close to those reported for the Merrill Lynch Index. Whereas low-grade issues promise to pay rates about 5 percentage points over 10-year Treasury yields, investors end up earning about 3 percentage points above the returns on Treasuries because of restructurings.

Overall, the data suggest that a risk premium of about 3 percentage points fairly compensates investors for the risk in holding low-grade bonds. The promised yield is set so that after taking account of the probability of default, investors expect to earn an average rate of return 3 percentage points in excess of the yield on 10-year Treasury bonds.

THE ROLE OF DIVERSIFICATION IN BOND INVESTING

High-grade bonds of a given maturity are basically all alike. Because the payments they offer are fixed in advance, the only thing that can cause the price to move is a change in the level of interest rates. When interest rates change, bond prices adjust so that the yield on the bonds remains equal to the level of interest rates in the market as described in our previous example. Because all high-grade bonds will adjust in the same way, the risk reduction benefits of diversification are minimal.

The same is not true of low-grade bonds. The returns on low-grade bonds depend not only on movements in interest rates but also on the financial performance of the issuer. One company may get in trouble and need to restructure, while another does well and makes all its payments as promised. If an investor holds a diversified portfolio of low-grade bonds, the bad results for some issues tend to be canceled by superior results for other issues. As we explain in the next chapter, this reduces the risk of holding low-grade bonds.

SUPERIOR RETURNS FROM LOW-GRADE BOND INVESTING

There is one added aspect of investing in low-grade bonds that Michael Milken, the so-called “junk bond king,” used to emphasize. Milken said: Suppose that you are a superior investor who can identify underpriced securities. If you find and buy an underpriced stock, you will not realize a big gain until the market recognizes the error of its ways and corrects the underpricing. As long as the stock stays underpriced, the investor does not make a superior return. Being right is not sufficient; the market has to come to realize that you are right. With a low-grade bond, on the other hand, being underpriced means the market has required a promised yield that is too large because it has overestimated the probability of restructuring. In your wisdom, you recognize that the company is not likely to restructure, and buy the bond. If you are right and the company does not restructure, you receive the promised payments, not the lower, market-expected payments. What is more, you get those higher payments whether or not the market ever comes to agree with you.

CONCEPTUAL FOUNDATION 3

The third conceptual foundation is that the ultimate goal of investing is to fund future consumption. Therefore, investment performance (returns) should be measured in real terms, not nominal terms. The dollar is not a good measure of value because it has been shrinking throughout most of American history.

Because sophisticated investors recognize that the dollar is shrinking, securities are priced so as to maintain the level of expected real returns. This is particularly clear in the case of fixed income securities which have predetermined promised yields. Those yields will move up and down with expected inflation so as to keep expected real returns relatively constant. The real return earned on a fixed income security depends on the rate of inflation that occurs during the life of the investment compared to the rate expected at the beginning.