This bond can be viewed as one bond or as a portfolio of three bonds—one-period, two-period, and three-period pure discount bonds paying $75, $75, and $1075, respectively. The price on this portfolio is given by Equation (20.2).

The price of the portfolio is, of course, the same as the price of the bond. The yield to maturity on the bond lies between the spot rates. Let us examine what this implies for yield curves of coupon bonds relative to yield curves of pure discount bonds. Consider a downward sloping yield curve. The spot rates associated with the earlier coupon payments are higher than the spot rate associated with the final maturity. Since the yield to maturity lies between these rates, the yield to maturity on the coupon bond lies above the spot rate associated with the final payment (see Figure 20.5). The higher the coupon payments, the greater the importance of earlier payments relative to the last payment and the more important the influence of earlier spot rates on the yield to maturity. Thus the higher the coupon payment, the greater the difference between the yield to maturity on the coupon paying bond and the spot rate on the final payment.

High coupon bonds Medium coupon bonds Low coupon bonds Pure discount bonds

High coupon bonds Medium coupon bonds Low coupon bonds Pure discount bonds


Figure 20.5 shows the plot of yield to maturity on coupon bonds compared to pure discount bonds. As just discussed, the greater the coupon, the greater the difference between yields to maturity and the spot rate of the final payment.

If the yield curve is upward sloping, then the yield to maturity on coupon bonds lies below the yield to maturity on discount bonds. The larger the coupon, the greater the difference between the yield on the coupon and noncoupon debt. Figure 20.6 plots the yield to maturity on bonds with various coupons with upward sloping yield curves.

A number of organizations examine yield curves on coupon paying debt. Pure discount debt for government bonds did not exist at all for bonds with maturities over one year until the 1980s. When pure discount debt for longer maturities was first offered, it was created by brokerage firms removing coupons from coupon bonds and selling them off separately. These instruments are not quite equivalent to pure discount government bonds, since they may be less marketable than when the government originally issued them, and there is some risk of the brokerage firm defaulting. Furthermore, even now there are not enough of them to allow accurate estimation of the yield curve. Most firms plot yield curves of coupon paying debt rather than go through the process of estimating the yield curve for pure discount debt using techniques discussed in the appendix at the end of the chapter. Examining Figures 20.5 and 20.6 shows that the general shape of the yield curve is preserved if the coupon rate on bonds of vatying maturities is the same. The problem is that they are not the same. Most bonds with intermediate maturity are long-term bonds that were issued several years before. For example, a bond with a 7-year maturity might be a 30-year bond issued 23 years ago. Interest rates change dramatically over time. Thus the coupon rate on bonds of different maturities is likely to be very different. A yield curve drawn from coupon paying bonds is likely to be a mixture of the yield curves shown in Figures 20.5 and 20.6. In this case, even the shape need not be preserved.

Organizations examine yield curves for investment decisions and for determining interest rates to be offered their customers. Using coupon bonds can lead to very misleading yield curves and incorrect decisions.

Summary of the Term Structure of Interest Rates We have shown how spot rates can be used to arrive at the correct price of any bond. To estimate spot rates one should use


the methodology outlined in the appendix at the end of this chapter. Spot rates are determined by current one-period rates, expectations about future one-period rates, theories of institutional behavior, and risk preferences. Although we have not attempted to find a categorical answer to which of these term structure theories is correct, we have provided you with enough information about the contrasting theories to give insight into the term structure of interest rates.

Default Risk

Unlike government bonds, for corporate bonds and municipal bonds there is a risk that the coupon or principal payments will not be met. For these bonds it is necessary to make a distinction between promised return and expected return. A bond could promise a return of 12%, but if there were some probability that the principal or coupon might not be paid, its expected return could be 10%. In addition, since there is risk associated with these bonds, investors should require that the expected return be greater than the return on a similar bond that is default free. These concepts are illustrated in Table 20.8.

We have referred to the difference between the promised return and the expected return as the default premium. The difference between the expected return and the return on a default-free instrument is the risk premium. The investor requires this extra return because of the chance that a particular bond selected may default, resulting in a very poor and probably negative return.

Three large investment services estimate the likelihood of default for most corporate bonds: Moody's, Standard & Poor's, and Fitch. The estimates from Moody's and from Standard & Poor's are widely available. Their services are similar in that they classify bonds by likelihood of loss. Likelihood of loss includes both the probability of a missed, delayed, or partial payment and the size of the loss if a loss occurs. For example, consider two bonds with the same probability of a missed principal payment. If one of them has significant odds of paying a substantia] portion of the principal payment if missed, while the odds are that the other will pay none, then the bond with the higher payment receives the higher rating. Bond rating services divide bonds into discrete classes. Table 20.9 shows Moody's classification of bonds and their discussion of what the various classifications mean.

Many organizations are restricted to buying bonds that have achieved at least a certain rating. These restrictions may be imposed by regulatory authority, by perception of legal requirements of prudent investment, or by organizational policy. In addition, many brokerage firms put together pools of bonds and then issue shares in these pools. These pools ate normally restricted to A-rated bonds or better. These restrictions suggest the possibility of a segmented market between higher rated bonds and lower rated bonds; however, we know of no conclusive evidence on this issue.

Table 20.8 Components of Interest Rates on Corporate Bonds

Default premium Risk premium

Return on default-free bonds

Table 20.9 Key to Moody's Corporate Ratings

Aaa Bonds which are rated Aaa are judged to be of the best quality. They carry the smallest degree of investment risk and are generally referred to as "gilt edge." Interest payments are protected by a large or by an exceptionally stable margin and principal is secure. While the various protective elements are likely to change, such changes as can be visualized are most unlikely to impair the fundamentally strong position of such issues.

\a Bonds which are rated Aa are judged to be of high quality by all standards. Together with the Aaa group they comprise what are generally known as high-grade bonds. They are rated lower than the best bonds because margins of protection may not be as large as in Aaa securities or fluctuation of protective elements may be of greater amplitude or there may be other elements present that make the long-term risks appear somewhat larger than in Aaa securities.

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

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

I!a Bonds which are rated Ba are judged to have speculative elements; their future cannot be considered as well assured. Often the protection of interest and principal payments may be very moderate and thereby not well safeguarded during both good and bad times over the future. Uncertainty of position characterizes bonds in this class.

B Bonds which are rated B generally lack characteristics of the desirable investment.

Assurance of interest and principal payments or of maintenance of other terms of the contract over any long period of time may be small.

Oaa Bonds which are rated Caa are of poor standing. Such issues may be in default or there may be present elements of danger with respect to principal or interest.

Ca Bonds which are rated Ca represent obligations which are speculative in a high degree. Such issues are often in default or have other marked shortcomings.

C Bonds which are rated C are the lowest rated class of bonds and issues so rated can be regarded as having extremely poor prospects of ever attaining any real investment standing.

Moody's and Standard & Poor's classifications can be duplicated fairly accurately by utilizing a weighted average of firm characteristics as follows. A number of firm characteristics ate hypothesized as influencing Moody's or Standard & Poor's classifications. These char-tcteristics usually include variables such as the amount of earnings compared to the interest payments, the variability of earnings, the amount of debt in the capital structure, the net worth, and the amount of short-term assets compared to short-term liabilities. Data on these variables are collected for a number of publicly traded bonds along with the classification of ¿•ach bond by one of the bond rating services. Mathematical techniques exist for finding the combination of firm variables that best duplicates the classification of the rating agency. The combination is best in the sense that it most accurately reproduces the ratings. Once the iest combination is determined, it is then tested using data on other publicly traded bonds to ice how well it classifies them. Accurate classification of 70-80% of the bonds is not uncommon, with most bonds being only one rating away from the published ratings.

Reproducing public ratings is useful in order that bonds not classified by the public rating services can be inexpensively and accurately classified. The most obvious utilization of this system is in classifying private placements. Banks and insurance companies lend money to firms directly. These private placements are usually loans to small- or medium-sized companies that wish to avoid the expenses of issuing publicly traded debt (e.g., SEC registration, brokerage costs). Analysts make judgments concerning the likelihood and size of loss, the appropriate interest rate on the potential loan, and the decision on whether to lend. When individual lending officers are judged in part by the volume of loans they make, they tend to be optimistic about the likelihood of the firm repaying the loan in the future. A scheme that fairly accurately reproduces public ratings is a check on this optimism. These schemes are frequently used to rate all loans under consideration. The analyst is then required to justify any difference in interest rates he or she wishes to offer compared to what is normal given the rating the bond receives.

Table 20.10 shows the default experience in recent years on average for junk or high-yield bonds. Junk bonds are bonds rated below Baa if using Moody's ratings or below BBB if using Standard & Poor's. The default experience for this category of bonds is significantly higher than for other bonds, averaging about 3% between 1978 and 2000.

Another way of examining the default experience is to examine it over the life of the bond. Table 20.11 shows the cumulative default experience for newly issued bonds in each year subsequent to issue. Thus, the 31.51%for CCC bonds implies that 31.51% of the bonds rated CCC defaulted in the first 10 years. The default experience over the life of the bond is quite substantial for low-rated bonds outstanding for a number of years. Note also that as discussed earlier, the default experience in the first year tends to be less than in subsequent years.

Table 20.10 Historical Default Rate-Low Rated, Straight Debt Only 1978-2000


Par Value Outstanding

Par Value Default

Default Rate

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