Let's go back to the 4%, 4-year bond that is priced at 99.342. The bond's yield-to-maturity statistic, which also is called the redemption yield, is 4.182%. That is not the only yield statistic that can be used to describe the investor's rate of return. Another is known as the current yield, but I've also seen it named the running yield and the income yield. It is the annual coupon payment divided by the price of the bond. Here the current yield is 4.0265%.

I've always imagined that the current yield is a statistic created by a former equity trader because it is analogous to a stock's dividend yield. However, it is severely lacking as a measure of the rate of return on a bond. The numerator is the sum of the periodic coupon payments over the year and neglects the time value of money. A 4% bond that pays its coupon interest quarterly would have the same current yield as this one that pays annually if it also were priced at 99.342. Moreover, the denominator neglects the inevitable pull-to-par effect that moves the price over time toward par value, assuming no change in the probability of default. If the investor planned to sell the bond after a year, a horizon yield would provide a much better estimate of the rate of return, even though that calculation would require projecting a sale price. Assuming the price remains stable over time, while perhaps reasonable for equity, doesn't make much sense for a bond priced at a premium or a discount.

Another statistic sometimes reported for a bond is the simple yield. This one is also called the Japanese simple yield because sometimes it is used to quote JGBs (Japanese government bonds). Imagine a bond analyst (in Tokyo) looking at the current yield and thinking, “I've got to fix that numerator.” To get a better projected rate of return for a buy-and-hold investor, add the straight-line amortization of the gain (from buying at a discount) or loss (from buying at a premium). The simple yield on this bond is 4.192%.

Investments textbooks back in the olden days (before financial calculators and spreadsheets) used to demonstrate how to approximate the yield to maturity. Remember that an internal rate of return has no closed-form equation and needs to be obtained by trial-and-error search. Imagine another (now very old) bond analyst looking at the simple yield and thinking, “I've got to fix that denominator.” To get an improved rate of return statistic, use the average of the current price and the redemption payment. The approximate yield turns out to be 4.178%.

Although the approximate yield is not reported in practice, it is used behind the scenes – better said, under the keypad of a financial calculator or buried in the programming of a spreadsheet. It can be written generally using our notation.

(3.8)

When you use a calculator or spreadsheet program to solve for the yield when PV is not equal to PV, the approximate yield is the starting place for the trial-and-error search process. Note that if PV = PV, the approximate yield reduces to just PMT/FV (i.e., the coupon rate).

Other yield statistics that might be used to summarize this 4%, 4-year bond involve converting the periodicity. Because this bond has annual coupon payments, the yield to maturity of 4.182% is an effective annual rate. Equation 1.10 can be used to convert to semiannually, quarterly, monthly, and daily compounded annual yields of 4.139%, 4.118%, 4.104%, and 4.097%, respectively. Here is the conversion from compounding annually to monthly.

Equation 2.3 can be used to show that there is not much difference between continuous and daily compounding. This is the conversion to compounding continuously from compounding monthly.

These yield-to-maturity statistics, regardless of the periodicity, are all stated in what is called street convention. That means we neglect the actual timing of cash flows in terms of weekends and holidays. For example, if the 4%, 4-year bond was purchased for settlement on December 15, 2010 (a Wednesday) and matures on December 15, 2014 (a Monday), we assume the investor received the intervening coupon payments on December 15, 2011 (a Thursday), 2012 (a Saturday), and 2013 (a Sunday). In reality, those last two payments would have been made on the next Monday. Even though the timing of the payment is delayed, the payment amount represents interest accrued through December 15.

In contrast to the simplifying street convention assumptions, the true yield statistic solves for the internal rate of return given the specific calendar dates for cash flows, based on some schedule of bank holidays (e.g., London, New York, Tokyo). Using these assumed dates, the true yield is the solution for true in this equation.

This calculation uses an actual/365 day-count format. For example, there actually were 733 days between purchase on Wednesday, December 15, 2010, and the second coupon payment on Monday, December 17, 2012. That includes the two additional days for delayed payment from Saturday to Monday, plus a leap day. Solving this on a spreadsheet gives a true yield of 4.179%, a little lower than the street convention yield because of the delay in the receipt of payment. In essence, the street convention yield entails solving for the internal rate of return using integers (1, 2, 3, and 4) for the exponents in the denominator whereas the true yield uses nonintegers (1, 2.008219, 3.005479, and 4.002740).

Maybe you are wondering if we really need all of these yield statistics for the bond. The answer lies in the difference between data and information. Figure 3.4 illustrates a way of differentiating these two. Think of information as a subset of data – there are lots and lots of data out there, but information is special: Information is data that can be used in making a decision. We're talking about bonds, so what matters is whether the yield statistic helps make a buy, hold, or sell decision. Surely this is up to each decision maker, and my perspective is that of a bond math teacher, not a bond investor or trader. But I just cannot see how the current yield, or the simple yield, or even the approximate yield help in decision making. So I classify them in Figure 3.4 as data about the bond but not as information.

On the other hand, the street convention yield to maturity can be used to compare bonds and does provide some information about the investor's possible rate of return. One of the themes of this chapter is to examine the yield-to-maturity statistic in detail and point out its limitations (for instance, if the investor does not intend to hold the bond to maturity or if the

FIGURE 3.4 Data vs. Information

probability of default is not zero). All yield statistics summarize cash flows; it's just that the street convention yield to maturity does so most consistently. Plus, it has a well-defined periodicity, and that is critical in understanding and comparing interest rates.

True yields are tough to classify. In Figure 3.4,1 call them data but not information, but I can see how some of you would argue that a true yield is the best datum of the lot. After all, it reflects the actual timing of cash flows rather than assumed timing. Maybe to you it is information used in decision making, but to me it is data overkill. There are so many more important things to deal with in fixed-income analysis than whether a coupon payment years into the future happens to be made on Saturday or the following Monday. For instance, at what rates will you be able to reinvest all of those coupon cash flows?

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