U.S. government Treasury bills involve only one payment from the Treasury to the investor at maturity. That is, their coupon rate is zero. They are thus an example of zero coupon bonds, bonds with no intermediate cash flows between issue date and maturity. The knowledge of the prices of zero coupon bonds allow us to determine the discount factor Z(t, T ), as described in the previous sections. More specifically, a government zero coupon bond at time t with maturity T has a price equal to
Pz(t, T ) = 100 × Z(t, T ) (2.12) The subscript “z” is a mnemonic term for “zero” in zero coupon bond.
COUPON BONDS 43
The Treasury issues zero coupon bonds with maturities up to only 52 weeks. For longer maturities, the Treasury issues securities that carry a coupon, that is, they also pay a sequence of cash flows (the coupons) between issue date and maturity, in addition to the final principal. In particular, the U.S. government issues Treasury notes, which are fixed income securities with maturity up to 10 years; Treasury bonds, which have maturities up to 30 years; and TIPS (Treasury Inflation Protected Securities), which have coupons that are not constant, but rather are linked to a recent inflation rate figure. We will talk about TIPS more exhaustively in Chapter 7. For now, we only consider Treasury notes and bonds.
For convenience, we refer to both types as coupon bonds.
2.4.1 From Zero Coupon Bonds to Coupon Bonds
In this section we establish a relation between the prices of zero coupon bonds and coupon bonds. This relation forms the basis of much of the analysis that follows in later chapters, and so it is particularly important.
First, note that a coupon bond can be represented by the sequence of its cash payments.
For instance, the 4.375% Treasury note issued on January 3, 2006 and with maturity of December 31, 2007, pays a cash flow of $2.1875 on June 30, 2006, December 31, 2006, and June 30, 2007, while it pays $102.1875 on December 31, 2007. Given the sequence of cash flows, which are certain in the sense that the U.S. Government is extremely unlikely to default, we could compute the value of the bond itself if we knew the discount factors Z(t, T ) to apply to each of the four dates. In fact, we can discount each future cash flow using its own discount factor, and sum the results.
Fact 2.8 Consider a coupon bond at time t with coupon rate c, maturity T and payment dates T1, T2,...,Tn = T . Let there be discount factors Z(t, Ti) for each date Ti. Then the value of the coupon bond can be computed as
Pc(t, Tn) = c× 100
The subscript “c” is a mnemonic device for “coupon” in coupon bond. Formula 2.14 shows that the coupon bond can be considered as a portfolio of zero coupon bonds.
EXAMPLE 2.7
Consider the 2-year note issued on t = January 3, 2006 discussed earlier. On this date, the 6-month, 1-year, 1.5-years, and 2-year discounts were Z(t, t + 0.5) = 0.97862, Z(t, t + 1) = 0.95718, Z(t, t + 1.5) = 0.936826 and Z(t, t + 2) = 0.91707.
44 BASICS OF FIXED INCOME SECURITIES
Therefore, the price of the note on that date was
Pc(t, Tn) = $2.1875 × 4 i=1
Z(t, t + 0.5 × i) + $100 × 0.91707 = $99.997,
which was indeed the issue price at t.
We can also represent the value of the coupon bond by using the semi-annual interest rate r2(t, Ti), where Ti, i = 1, ..., n, are the coupon payment dates. This representation is derived from the basic one above, but it can be useful nonetheless to report it:
Pc(t, Tn) = A useful fact is the following:
Fact 2.9 Let the semi-annual discount rate be constant across maturities, r2(t, Ti) = r2
for every Ti. At issue date t = 0, the price of a coupon bond with coupon rate equal to the constant semi-annual rate c = r2is equal to par.
Pc(0, Tn) =
To understand the above fact, consider a 1-year note. Then, we can write Pc(0, T2) = c/2 × 100
This argument can be extended to many periods. The intuition is that any additional periods increase the cash flow by c/2 while they also increases the discount rate by the same amount r2/2. The two forces move in opposite directions (more cash flows imply higher prices, while the additional discount imply lower price).
2.4.1.1 A No Arbitrage Argument We can establish Equation 2.13 also by appeal-ing to a no arbitrage argument. In well-functionappeal-ing markets in which both the coupon bond Pc(t, Tn) and the zero coupon bonds Pz(t, Ti) are traded in the market, if Equation 2.13 did not hold, an arbitrageur could make large risk-free profits. For instance, if
Pc(t, Tn) < c
COUPON BONDS 45
then the arbitrageur can buy the coupon bond for Pc(t, Tn) and sell immediately c/2 units of zero coupon bonds with maturities T1, T2,..,Tn−1and (1 + c/2) of the zero coupon bond with maturity Tn. This strategy yields an inflow of money to the arbitrageur that is equal to the difference between the right-hand side and the left-hand side of Equation 2.22. At every other maturity Tithe arbitrageur has a zero net position, as he receives the coupon from the Treasury and turns it around to the investors to whom the arbitrageur sold the individual zero coupon bonds. We note that this reasoning is the one that stands behind the law of one price, introduced in Fact 1.1 in Chapter 1, the fact that securities with identical cash flows should have the same price. The following example further illustrates the concept.
EXAMPLE 2.8
In Example 2.7, suppose that the 2-year note was trading at $98. An arbitrageur could purchase, say, $98 million of the 2-year note, and sell $2.1875 million of the 6-month, 1-year and 1.5-year zero coupons, and $102.1875 million of the 2-year zero coupon bond. The total value of the zeros the arbitrageur sells is $99.997 million, realizing approximately $2 million. The strategy is risk free, because at each coupon date in the future, the arbitrageur receives $2.1875 million from the Treasury, which he simply turns over to the investors who bought the zero coupon bonds. Similarly, at maturity, the arbitrageur receives $102.1875 million from the Treasury, and again turns it around to the investors of the last coupon.
In well-functioning markets such arbitrage opportunities cannot last for long. Thus, Equa-tion 2.13 should hold “most of the time”. It may happen that due to lack of liquidity or trading, some arbitrage opportunities may be detectable in the relative pricing of zero coupon bonds, such as STRIPS, and coupon bonds. However, these arbitrage opportunities are seldom exploitable: As soon as an arbitrageur tries to set up an arbitrage like the one described above, prices move instantly and the profit vanishes. Because expert arbitrageurs know this fact, some apparent mispricing may persist in the market place. We will regard such situations as “noise”, that is, a little imprecision in market prices due to liquidity or external factors that sometimes impede the smooth functioning of capital markets.
2.4.2 From Coupon Bonds to Zero Coupon Bonds
We can also go the other way around: If we have enough coupon bonds, we can compute the implicit value of zero coupon bonds from the prices of coupon bonds. Equation 2.13 be used to estimate the discount factors Z(t, T ) for every maturity. The following example illustrates the reasoning:
EXAMPLE 2.9
On t = June 30, 2005, the 6-month Treasury bill, expiring on T1 = December 29, 2005, was trading at $98.3607. On the same date, the 1 year to maturity, 2.75%
Treasury note, was trading at $99.2343. The maturity of the latter Treasury note is T2 = June 30, 2006. Given Equation 2.13, we can write the value of the two
46 BASICS OF FIXED INCOME SECURITIES
securities as:7
Pbill(t, T1) = $98.3607 = $100 × Z(t, T1) (2.23) Pnote(t, T2) = $99.2343 = $1.375 × Z(t, T1) + $101.375 × Z(t, T2) (2.24) We have two equations in two unknowns [the discount factors Z(t, T1) and Z(t, T2)].
As in Section 2.1, from the first equation we obtain the discount factor Z(t, T1) =
$98.3607/$100 = 0.983607. We can substitute this value into the second equation, and solve for Z(t, T2) to obtain:
Z(t, T2) = $99.2343 − $1.375 × Z(t, T1)
$101.375 =$99.2343 − $1.375 × 0.983607
$101.375 = 0.965542
The prices of coupon bonds, then, implicitly contain the information about the market time value of money. This procedure can be iterated forward to obtain additional terms.
EXAMPLE 2.10
On the same date, t = June 30, 2005, the December 31, 2006 Treasury note, with coupon of 3%, was trading at $99.1093. Denoting by T3= December 31, 2006, the price of this note can be written as
Pnote(t, T3) = $99.1093 = $1.5 × Z(t, T1) + $1.5 × Z(t, T2) + $101.5 × Z(t, T3) (2.25) We already determined Z(t, T1) = 0.983607 and Z(t, T2) = 0.965542 in Example 2.9. In Equation 2.25 the only unknown element is Z(t, T3). This is one equation in one unknown, and so we can solve for the Z(t, T3) to obtain
Z(t, T3) = $99.1093 − $1.5 × (Z(t, T1) + Z(t, T2))
$101.5
= $99.1093 − $1.5 × (0.983607 + 0.965542)
$101.5
= 0.947641
If we have a sufficient amount of data, we can proceed in this fashion for every maturity, and obtain all of the discount factors Z(t, T ). This methodology is called the bootstrap methodology.
Definition 2.5 Let t be a given date. Let there be n coupon bonds, with coupons ci, maturities Ti and prices denoted by P (t, Ti). Assume that maturities are at regular intervals of six months, that is, T1 = t + 0.5 and Ti = Ti−1+ 0.5. Then, the bootstrap methodology to estimate discount factors Z(t, Ti) for every i = 1, ..., n is as follows:
1. The first discount factor Z(t, T1) is given by Z(t, T1) = Pc(t, T1)
100 × (1 + c1/2) (2.26)
7Notice a little approximation in this computation: The T-note would pay its coupon on December 31, 2005, rather than December 29. We assume that both dates correspond, approximately, to T1.
COUPON BONDS 47
2. Any other discount factor Z(t, Ti) for i = 2, ..., n is given by
Z(t, Ti) = Pc(t, Ti) − ci/2 × 100 × i−1
j=1Z(t, Tj))
100 × (1 + ci/2) (2.27)
This procedure is relatively simple to implement, as the example above shows. One of the issues, though, is that bond data at six-month intervals are not always available.
Unfortunately, this procedure requires all of the bonds, because otherwise the iterative procedure stops and there is no way to keep going. The appendix at the end of this chapter reviews some other methodologies that are widely used to estimate the discount factors Z(0, T ) from coupon bonds.
2.4.3 Expected Return and the Yield to Maturity
How can we measure the expected return on an investment in Treasury securities? As-suming the investor will hold the bond until maturity, computing the expected return on an investment in a zero coupon bond is relatively straightforward, as the final payoff is known and there are no intermediate cash flows. Thus, quite immediately, we have
Return on zero coupon bond = 1
Z(t, T ) −1 (2.28)
This is the return between t and T . It is customary to annualize this amount, so that
Annualized return on zero coupon bond =
1
Z(t, T )
1
T −t − 1 (2.29)
This, of course, corresponds to the annually compounded yield on the zero coupon, as in Equation 2.6 for n = 1.
For coupon bonds it is more complicated. A popular measure of return on investment for coupon bonds is called yield to maturity, which is defined as follows:
Definition 2.6 Let Pc(t, T ) be the price at time t of a Treasury bond with coupon c and maturity T . Let Tidenote the coupon payments times, for i = 1, ..., n. The semi-annually compounded yield to maturity, or internal rate of return, is defined as the constant rate y that makes the discounted present value of the bond future cash flows equal to its price.
That is, y is defined by the equation
Pc(t, T ) = n i=1
c/2 × 100
(1 + y/2)2×(Ti−t) + 100
(1 + y/2)2×(Tn−t) (2.30)
Before moving to interpret this measure of return on investment, it is important to recognize a major distinction between the formula in Equation 2.30 and the one that we obtained earlier in terms of discount factors, namely Equation 2.15. Although they appear the same, it is crucial to note that the yield to maturity y is defined as the particular constant rate that makes the right-hand side of Equation 2.30 equal to the price of the bond. Instead,
48 BASICS OF FIXED INCOME SECURITIES
Equation 2.15 is the one defining the price of the bond from the discount factors Z(t, T ).
Unless the term structure of interest rates is exactly flat, the yields at various maturities are different, and will not coincide with the yield to maturity y. Indeed, to some extent, the yield to maturity y can be considered an average of the semi-annually compounded spot rates r2(0, T ), which define the discount Z(0, T ). However, it is important to note that this
“average” depends on the coupon level c. In fact:
Fact 2.10 Two different bonds that have the same maturity but different coupon rates c