Zero-coupon security

As discussed in chapter 1, there are two types of fixed-income securities zero-coupon bonds, also known as discount bonds or strips, and coupon bonds. A zero-coupon bond makes a single payment on its maturity date, while a coupon bond makes interest payments at regular dates up to and including its maturity date. A coupon bond may be regarded as a set of strips, with the payment on each coupon date and at maturity being equivalent to a zeto-coupon bond maturing on that date. This equivalence is not purely academic. Before the advent of the formal market in U.S. Treasury strips, a number of investment banks traded the cash flows of Treasury securities as separate zero-coupon securities. 

Between the short- and long-term horizon dates is one at which the net effect of the change in reinvestment rate on the bond s future value is close to zero. At this date, the bond behaves like a single-cash-flow or zero-coupon security, and its future value can be predicted with greater certainty, no matter what the yield curve does after its purchase. Defining this date as Sh interest periods after the purchase date and Ph as the value of the bond at that point, it can be shown that the bond s rate of return up to this horizon date is the value for rntn that solves equation (16.6). 

As noted in chapter 2, a Treasury bond can be seen as a bundle of individual zero-coupon securities, each maturing on one of the bond s cash flow payment dates. In this view, the Treasury s price is the sum of the present values of all the constituent zero-coupon bond yields. Assume that the spot rates for the relevant maturities—ri,r2,rg,.rj f—can be observed. If a bond pays a semiannual coupon computed at an annual rate of C from period 1 to period N, its present value can be derived using equation (16.7). 

Because it is affected by current demand, the yield of a particular zero-coupon bond at any time may differ from the equivalent-maturity spot yield. When investors value an individual zero-coupon bond less highly as a stripped security than as part of a coupon bond s theoretical package of zero-coupon cash flows, the strip s yield will be above the spot rate for the same maturity. The opposite happens when investors prefer to hold the zero-coupon security. 

The simplest approach assumes, somewhat unrealistically, that the yield curve is flat and moves only in parallel shifts, up or down. It considers a bond to be a package of zero-coupon securities whose values are discounted and added together to give its theoretical price. The advantage of this approach is that each cash flow is discounted at the interest rate for the relevant term, rather than at a single internal rate of return, as in the conventional approach. Given the flat yield curve, however, this approach reduces to (17.3). An example of its application follows. 

DIHBir 1.4 Bloomberg Security Description Screen for a Zero-Coupon Bond Issued by BNP Paribus... 

The source of dollar return called reinvestment income represents the interest earned from reinvesting the bond s interim cash flows (interest and/or principal payments) until the bond is removed from the investor s portfolio. With the exception of zero-coupon bonds, fixed income securities deliver coupon payments that can be reinvested. Moreover, amortizing securities (e.g., mortgage-backed and asset-backed securities) make periodic principal repayments which can also be invested. 

As noted above, the bond market includes securities, known as zero-coupon bonds, or strips, that do not pay coupons. These are priced by setting C to 0 in the pricing equation. The only cash flow is the maturity payment, resulting in formula (1.18)... 

A zero-coupon bond is the simplest fixed-income security. It makes no coupon payments during its lifetime. Instead, it is a discount instrument, issued at a price that is below the face, or principal, amount. The rate earned on a zero-coupon bond is also referred to as the spot interest rate. The notation P t, T) denotes the price at time r of a discount bond that matures at time T, where T >t - The bond s term to maturity, T - t, is... 

In the academic literature, the risk-neutral price of a zero-coupon bond is expressed in terms of the evolution of the short-term interest rate, r t)—the rate earned on a money market account or on a short-dated risk-free security such as the T-bill—which is assumed to be continuously compounded. These assumptions make the mathematical treatment simpler. Consider a zero-coupon bond that makes one payment, of 1, on its maturity date T. Its value at time ris given by equation (3.14), which is the redemption value of 1 divided by the value of the money market account, given by (3.12). 

Market practitioners armed with a term-structure model next need to determine how this relates to the fluctuation of security prices that are related to interest rates. Most commonly, this means determining how the price T of a zero-coupon bond moves as the short rate r varies over time. The formula used for this determination is known as Itos lemma. It transforms the equation describing the dynamics of the bond price P into the stochastic process (4.5). 

In the United States, Canada, and New Zealand, indexed bonds can be stripped, allowing coupon and principal cash flows to be traded separately. This obviates the need for specific issues of zero-coupon indexed securities, since the market can create products such as deferred-payment indexed bonds in response to specific investor demand. In markets allowing stripping of indexed government bonds, a strip is simply a single cash flow with an inflation adjustment. An exception to this is in New Zealand, where the cash flows are separated into three components the principal, the principal inflation adjustment, and the inflation-linked coupons—the latter being an indexed annuity. 

Synthetic convertible notes are securities with fixed coupons, typically set at a relatively low level, whose total return is linked to an external source, such as the level of an equity index or the price of a specific security. In one common structure, the note is redeemable above par if the reference index or security value exceeds a stated minimum. The notes thus give investors the opportunity to profit from the benchmarks performance while providing the safety net of redemption at par should this performance fall short. Another typical synthetic convertible structure is the zero-coupon note. These notes are issued at par and redeemable at par, or higher, if a specified equity index performs better than a stated level. 

Spot yields cannot be directly observed in the market. They can, however, be computed from the observed prices of zero-coupon bonds, or strips, if a liquid market exists in these securities. An implied spot yield curve can also, as the previous section showed, be derived from coupon bonds prices and redemption yields. This section explores how the implied and actual strip yields relate to each other. 

This section discusses the factors that must be assessed in analyzing the relative values of government bonds. Since these securities involve no credit risk (unless they are emerging-market debt), credit spreads are not among the considerations. The zero-coupon yield curve provides the framework for all the analyses explored. 

A bond may be valued relative to comparable securities or against the par or zero-coupon yield curve. The first method is more appropriate in certain situations. It is suitable, for instance, when a low-coupon bond is trading rich to the curve but fair compared with other low-coupon bonds. This may indicate that the overpricing is a property not of the individual bond but of all low-coupon bonds. 

Zero-coupon indexation. Zero-coupon indexed bonds have been issued in Sweden. As their name implies, they pay no coupons the entire inflation adjustment occurs at maturity, applied to their redemption value. These bonds have the longest duration of all indexed securities and no reinvestment risk. 

Equation (17.7) differs from the conventional redemption yield formula in that every cash flow is discounted, not by a single rate, but by the zero-coupon rate corresponding to the maturity period of the cash flow. To apply this equation, the zero-coupon-rate term structure must be known. These rates, however, are not always readily observable. Treasury prices, on the other hand, are and can be used to derive implied spot interest rates. (Although in the market the terms are used interchangeably, from this point on, zero coupon will be used only of observable rates and spot only of derived ones.) To see how the derivation works, consider the 10 hypothetical U.S. Treasuries whose maturities, prices, and yields are shown in FIGURE 17.9. Assume that the yield curve is positive and that the securities settlement date—March 1, 1999—is a coupon date, so none of them has accrued interest. 

Treasury notes have maturities of between 2 and 10 years. Because of their longer maturities, these notes have more interest rate risk associated with them and so their prices fluctuate more than T-bill prices. The U.S. government used to issue Treasury bonds, which carried maturities of 15, 20, and 30 years. The 30-year Treasury bond was just retired in November 2001. Another Treasury security is a "strip" or zero-coupon Treasury security created by separating the income streams of coupon payments and principal, wherein the holder receives no coupon payments, buys the bond at a discount, and is returned the principal at par. There is a high degree of volatility associated with strips. 

Non-interest-beanng securities are also referred to as discounted securities. Unlike regular corporate bonds, which pay periodic interest (i.e., pay a coupon), interest is earned on these bonds by their appreciation in price over time. That is, these securities originally sell for less than their maturity or face value. All other factors being constant, the price approaches the face value as the time to maturity approaches zero. Note that this is not the same thing as a coupon bond that is selling at a discount. 

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