It's finally time to get realistic about bond prices and yields. So far in this chapter I've conveniently assumed exactly four years to maturity on the bonds in the examples and that the next coupon payment is due one year into the future. That simplifies the calculations to draw your attention to the factors impacting yields and (pretax) rates of return – coupon reinvestment rates and the probability of default. You've probably heard the old saying “Even a broken clock gives the correct time twice a day.” The analogy is that knowing how to do bond math on coupon dates for standard semiannual payment corporate and Treasury bonds makes you a valuable person – twice a year.

Extending basic bond math to between coupon dates is not hard conceptually, although some real-world accommodations are made. Suppose that the current coupon period covers T days and that the bond is being priced for settlement t days into the period. Therefore, t/T is the fraction of the period that has gone by and 1 – t/T is the fraction that remains. Here is a general version of equation 3.4, discounting the coupon payments (PMT) and principal redemption (FV) over the remaining N payments at the yield to maturity per period (y).

(3.9)

On the right side of equation 3.9, the next coupon payment is discounted back over the fraction of the period (1 – t/T) until that cash flow is

received; the following payment adds a full period to that fraction (2 – t/T), and so forth. The left side is the sum of the present values of the cash flows and is the full price for the bond on the settlement date. That full price, which often is called the dirty or invoice price in practice, is decomposed into the flat price (Flat), which also is called the clean price, and the accrued interest (AI). Why dirty and clean? Surely it's not true, but I like to say it's because accrued interest in practice is impure because it's not theoretically correct.

Accrued interest is the compensation to the seller of the bond for interest income since the last coupon date. It is calculated as a straight-line share of the forthcoming payment. It is the fraction of the period that has elapsed times the amount of the payment.

(3.10)

Determining the fraction t/T is not an obvious matter because it depends on the day-count convention specified in the bond's documentation. Government bonds in the U.S. use actual/actual, whereas corporate, agency, and municipal bonds use 30/360. As we saw with money market instruments in Chapter 1, there are other possibilities – actual/360 and actual/365 – but these are not commonly used with bonds. Also, note that there are various versions of 30/360. For instance, how many days are between April 1 and May 31 using a 30/360-day count? The answer is either 60 or 61, depending on the specification.

Multiply the numerator and denominator by (1 + y)t/T in equation 3.9 and substitute in equations 3.5 and 3.10 to get a general closed-form relationship between the present and future cash flows and the yield-to-maturity statistic.

(3.11)

This equation can be used to solve for the street convention yield to maturity because the key assumption that payments are made on calendar dates without regard to weekends and holidays means that N is an integer. An important point is that the same fraction t/T is used on both sides of the equation whether the bond is a Treasury using the actual/actual day-count convention or a corporate using 30/360.

Another yield statistic you'll see quoted on corporate bonds is the U.S. government equivalent. The idea is to recalibrate the yield using an actual/actual day-count convention instead of 30/360. An example will clarify this. Let's assume that an 8%, semiannual payment high-yield corporate bond was being priced to yield 8.00% (s.a.) for settlement on Valentine's Day, February 14, 2011. The bond was issued at par value on November 15, 2010, and matures on November 15, 2020, a Sunday, which we'll ignore because the yield is stated in street convention. The next coupon was due on May 15, 2011, also a Sunday, which also is ignored.

Substitute into equation 3.11 t/T = 89/180 (the corporate bond uses a 30/360 day-count for accrued interest), PMT = 4 (percent of par value), y = 0.04 (the yield to maturity per semiannual period), TV = 100 (percent of par value), and N = 20 (semiannual periods to maturity as of the beginning of the period).

The accrued interest is 1.977778 [= 89/180 * 4]; the flat (or clean) price is 99.980394; and the full (or dirty) price is 101.958172 [= 99.980394 + 1.977778], all stated as a percentage of par value. You've noticed, of course, that the price on this high-yield corporate bond is a little below par value even though the coupon rate and yield to maturity are equal.

This example breaks the bond pricing rule you no doubt remembered – “If the coupon rate equals the yield to maturity, the bond is priced at par value” – because we are now between coupon dates. Think of it this way: If you buy the bond, you need to compensate the seller for interest earned during the fraction of the period (89/180) that he or she has owned the bond. But, to be theoretically correct, you need to include in the purchase price only the present value of the accrued interest – that is, the present value of 1.977778. That's because the new owner of the bond will receive the next coupon payment on May 15, 2011. Market practice neglects that time-value-of-money concern, so the flat price registers that neglect. Accountants always have to balance materiality and practicality with theoretical correctness. In any case, the total price of 101.958172 (percent of par value) is correct in that it is the present value of the cash flows discounted at the yield to maturity.

The U.S. government equivalent yield for this bond turns out to be 8.0050% (s.a.), slightly above the street convention yield. It is the solution for the yield to maturity using 91/181 on the right side of the equation instead of 89/180. That's because the fraction of the period gone by on an actual/actual basis is 91/181. Notice that the left side of the equation is the same because the purchase price is what it is.

Is the U.S. government equivalent yield information or just more data? I think it can be useful information if you are calculating this corporate bond's spread over a benchmark Treasury bond. Then the two yield statistics are on a fully comparable basis.

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