BOND PRICES AND YIELDS TO MATURITY IN A WORLD OF NO ARBITRAGE
Modern financial theory of bond pricing rests on the principle of no arbitrage. Some call this the law of one price or, even more grandly, the fundamental theorem of finance. It essentially is a theory of relative prices – the idea is that if we observe prices on some fundamental building blocks, for instance, zero-coupon bonds, we can deduce the fair prices on coupon bonds that promise the same future cash flows. We don't try to figure out the demand and supply for each bond in the marketplace. Instead we observe the market prices on the most actively traded securities and value the remainder assuming arbitrage opportunities are exploited and priced away.
No-arbitrage pricing is a powerful argument in developed financial markets because it does not require a lot of assumptions about information, behavior, risk aversion, and expectations. All we need are motivated and capitalized traders at hedge funds and proprietary trading desks at financial institutions. If there is an arbitrage opportunity out there, they can be trusted to find it and execute the trades to capture the profit. If you have studied the Capital Asset Pricing Model (the famous CAPM), you might recall the long list of assumptions needed to get those powerful, but theoretical, results about equity prices and the market price of risk. No-arbitrage pricing is far less ambitious – it just needs a starting place of some actively traded securities.
It's common in no-arbitrage valuation to neglect transactions costs and to assume that arbitrageurs actually can carry out the requisite trades. These assumptions hold up pretty well in the U.S. Treasury market because of the large supply of coupon bonds and zeros (i.e., C-STRIPS and P-STRIPS). In principle, arbitrageurs could use special-purpose entities to create zero-coupon corporate bonds, as in the TIGRS, CATS, and LIONS story in Chapter 2, but then the transactions costs would not be trivial.
Those caveats aside, let's now determine the no-arbitrage price for a 4-year, 4% annual coupon payment corporate bond in three different scenarios, neglecting transactions costs. First, assume that the sequence of zero-coupon bond yields is 3.500%, 3.800%, 4.100%, and 4.200%, an upward-sloping spot curve. (“Spot rate” is a commonly used synonym for zero-coupon rate.) Discounting each of the cash flows at the corresponding spot rate produces a bond price of 99.342 (percent of par value).
TABLE 3.1 A Yield-to-Maturity and Several Zero-Coupon Curves
Date |
Payment |
Zero Rate (%) |
PV |
Zero Rate (%) |
PV |
Zero Rate (%) |
PV |
Zero Rate (%) |
PV |
1 |
4 |
3.500 |
3.865 |
4.920 |
3.812 |
2.130 |
3.917 |
4.182 |
3.839 |
2 |
4 |
3.800 |
3.712 |
4.690 |
3.650 |
2.820 |
3.784 |
4.182 |
3.685 |
3 |
4 |
4.100 |
3.546 |
4.300 |
3.525 |
3.650 |
3.592 |
4.182 |
3.537 |
4 |
104 |
4.200 |
88.219 |
4.160 |
88.355 |
4.250 |
88.050 |
4.182 |
88.280 |
PRICE |
99.342 |
99.342 |
99.342 |
99.342 |
|||||
YTM |
4.1820% |
4.1820% |
4.1820% |
4.1820% |
Second, assume that the zero-coupon rates instead are 4.920%, 4.690%, 4.300%, and 4.160%, a downward-sloping spot curve. It's no surprise to those who see where this is going that the bond price turns out once again to be 99.342.
Third, to complete the example, suppose that the zero-coupon yield curve is remarkably flat at 4.182%.
Table 3.1 shows output from the spreadsheet that I used to concoct these examples by trial and error, including a set of spot rates left out of the text. Note that the present value of each future cash flow differs but all sum to 99.342.
These examples suggest that the starting place in no-arbitrage bond valuation is the zero-coupon (or spot) yield curve and these rates are used to value coupon bonds. In practice, it goes the other way. The most actively traded securities are the newly issued coupon bonds for standard maturities – for instance, 3-month, 6-month, 1-year, 2-year, 3-year, 5-year, 7-year, 10-year, and 30-year Treasuries. Then from the prices and coupon rates on these securities, we deduce the no-arbitrage yields on zero-coupon bonds. We work through this “bootstrapping” technique in detail in Chapter 5.
Let's assume that this 4-year, 4% annual payment corporate bond is priced at 99.342 (percent of par value). Notice that if we included transaction costs for buying and selling zero-coupon bonds, we would not be able to give such an exact no-arbitrage value to the bond. Instead, we would have a range of prices, as in a typical bid-ask spread. But given a particular bond price, the yield to maturity is the internal rate of return (IRR) on the cash flows. An IRR is the uniform discount rate such that the sum of the present values of the future cash flows discounted at that particular interest rate for each time period equals the price of the bond. Obviously, as we can tell by the third scenario, the IRR for this bond is 4.182%.
I've come up with this example to make an important point and correct a statement that is often made about bond yields to maturity. You might read or hear that the problem with using the yield-to-maturity (YTM) statistic on a coupon bond is that it assumes a flat yield curve. But now we see that simply is not true. The first scenario has an upward-sloping zero-coupon curve and gets a no-arbitrage value of 99.342. The second has a down- sloping curve and gets the same price – a price that corresponds to the yield to maturity of 4.182% on the 4-year, 4% bond.
In my opinion, the correct way to think about a yield to maturity is as a summary statistic about the cash flows on the bond. It is a weighted average of the sequence of zero-coupon rates with most of the weight on the last cash flow because that one includes the principal. So, 4.182% is an “average” of 3.50%, 3.80%, 4.10%, and 4.20%, just as it is an “average” of 4.92%, 4.69%, 4.30%, and 4.16%. The quotes around “average” remind us that it is not a simple average found by adding the rates and dividing by four. Instead, it is a “present value average” of the spot rates in the sense that it obtains the same price for the bond.
The key point is that the yield-to-maturity statistic boils the many cash flows down into a single number that might be useful in making a decision regarding the bond, with emphasis on “might be.” Additional inputs usually are needed to make a buy, hold, or sell decision – tax rates, expected coupon reinvestment rates, and the default probability. Remember the old quip about “a nonswimmer drowning trying to cross a river that has an average depth of 12 inches.” Sometimes averages are insightful summary statistics (as in baseball), sometimes they are not (as in football, in my opinion). In sum, the yield to maturity on a fixed-income bond does not presume a flat yield curve.
Equation 3.3 presents the generalization of the bond pricing equation.
(3.3)
where PV is the no-arbitrage value of the N-period bond – the sum of the present values of the cash flows, each of which is discounted using the zero-coupon rate that corresponds to the period (z1, z2,.. ., zN), PMT is the coupon payment per period, and FV is the principal (usually taken to be 100 so the price can be interpreted as the percentage of par value). In Chapter 2 we used Years * PER for the time until maturity, where PER is the periodicity (number of periods in the year), and a discount rate of APRPer/PER. Now we focus on the periodic cash flows and rates. Later in the chapter we deal with accrued interest and pricing the bond for settlement between coupon dates.
The yield to maturity (y) per period is the internal rate of return given the cash flows.
(3.4)
Comparing equations 3.3 and 3.4, we see why the bond yield can be interpreted as a “weighted average” of the zero-coupon rates – the PV, PMT, FV, and N are the same. Doing some algebra (i.e., taking the sum of the finite geometric series) allows this rearrangement. The steps are shown in the Technical Appendix.
(3.5)
Equation 3.5 is programmed into financial calculators for time-value- of-money problems. But first a couple of changes are usually made.
(3.6)
In equation 3.6, the interest rate per period is divided by 100 so that it can be entered as a percentage, not as a decimal. Also, the sum of the three terms is zero so that at least one of PV, PMT, or FV must be entered as a negative. That allows for the interpretation that negative inputs imply cash outflows and positive inputs are inflows. I've found that for bond math calculations, it's best to use PV as negative and PMT and FV as positive, thereby taking the perspective of the fixed-income investor.
An algebraic rearrangement of 3.5 is shown in equation 3.7.
(3.7)
Here c is the coupon rate per period, PMT/FV. This expression indicates the connection between the price of the bond vis-a-vis par value and the coupon rate vis-a-vis the yield to maturity. These are the well-known (and well-remembered) rules: (1) If the bond is priced at par value (PV = FV), the coupon rate and the yield to maturity are equal (c = y); (2) if the price is a discount below par value (PV < FV), the coupon rate is less than the yield (c < y); and (3) if the price is a premium above par value (PV > FV), the coupon rate is greater than the yield (c > y). These rules apply to a coupon payment date when N is an integer. It will have to be revised slightly for settlement dates between coupon payments – more on that later in the chapter.
At this point I cannot resist relating bond pricing to assessing a person's quality of life. Think of the coupon rate as what you're promised to get in life (assuming the “issuer” doesn't default) and the yield to maturity as what you really need (to pay full par value). So, if you are getting more than you need, your life is trading at a premium. But if you are not getting what you need, your life is priced at the discount. Remember that if you have been dealt a low coupon rate, you still can have a premium life – it's a matter of keeping your needs under control. Okay, enough bond math philosophy.