Zero-coupon bond market

In order to calculate the range of implied forward rates, we require the term stmcture of spot rates for all periods along the continuous discount function. This is not possible in practice, because a bond market will only contain a finite number of coupon-bearing bonds maturing on discrete dates. While the coupon yield curve can be observed, we are then required to fit the observed curve to a continuous term structure. Note that in the United Kingdom gilt market, for example there is a zero-coupon bond market, so that it is possible to observe spot rates directly, but for reasons of liquidity, analysts prefer to use a fitted yield curve (the theoretical curve) and compare this to the observed curve. 

Although the term zero-coupon rate refers to the interest rate on a discount instrument that pays no coupon and has one cash flow at maturity, constructing a zero-coupon yield curve does not require a functioning zero-coupon bond market. Most financial pricing models use a combination of the following instruments to construct zero-coupon yield curves ... 

Bond market participants take a keen interest in both the cash and the zero-coupon (spot) yield curves. In markets where an active zero-coupon bond market exists, the spreads between implied and actual zero-coupon yields also receive much artention. 

In this thesis we derived new methods for the pricing of fixed income derivatives, especially for zero-coupon bond options (caps/floor) and coupon bond options (swaptions). These options are the most widely traded interest rate derivatives. In general caps/floors can be seen as a portfolio of zero-coupon bond options, whereas a swaption effectively equals an option on a coupon bond (see chapter (2)). The market of these LIBOR-based interest rate derivatives is tremendous (more than 10 trillion USD in notional value) and therefore accurate and efficient pricing methods are of enormous practical importance. 

We first set the scene by introducing the interest-rate market. The price of a zero-coupon bond of maturity T at time t is denoted by P(t, T) so that its price at time 0 is denoted by / (O, T). The process followed by the bond price is a stochastic one and therefore can be modelled equally, options that have been written on the bond can be hedged by it. If market interest rates are constant, the price of the bond at time t is given by This enables us to state that given... 

In this section, we describe the relationship between the price of a zero-coupon bond and spot and forward rates. We assume a risk-free zero-coupon bond of nominal value 1, priced at time t and maturing at time T. We also assume a money market bank account of initial value P t, T) invested at time t. The money market account is denoted M. The price of the bond at time t is denoted P t, T) and if today is time 0 (so that t > 0), then the bmid price today is unknown and a random factor (similar to a future interest rate). The bond price can be related to the spot rate or forward rate that is in force at time t. 

The continuously compounded constant spot rate is r as before. An investor has a choice of purchasing the zero-coupon bond at price P(t, T), which will return the sum of 1 at time T, or of investing this same amount of cash in the money market account, and this sum would have grown to 1 at time T. We know that the value of the money market accoxmt is given by Me If M must have a... 

It might be considered to be more reahstic to consider that there are no constant parameters for the drift rate and the standard deviation that would ensure that the price of a zero-coupon bond at any time is exactly the same as that suggested by observed market yields. For this reason, a modified version of the Vasicek model has been described by Hull and White (1990), known as the Hull-White or extended Vasicek model, which we will consider later. 

The expression for the value of the money market account can be used to determine the expression for the zero-coupon bond price, which we denote as P(t, T). The money market account earns interest at the spot rate rit), while the bond price is the present value of 1 discounted at this rate. Therefore, the inverse of Equation (4.8) is required, which is... 

The no-arbitrage condition is set by defining the price of a zero-coupon bond that matures at time T in terms of an accumulation factor B(t) which is the value of a money market account that is invested at time 0 and reinvested at time t at an interest rate of r(t). This accumulation factor is defined as Equation (4.31) ... 

To obtain the price of an inflation-linked bond, it is necessary to determine the value of coupon payments and principal repayment. Inflation-linked bonds can be structured with a different cash flow indexation. As noted above, duration, tax treatment and reinvestment risk, are the main factors that affect the instrument design. For instance, index-aimuity bmids that give to the investor a fixed annuity payment and a variable element to compensate the inflation have the shortest duration and the highest reinvestment risk of aU inflation-linked bonds. Conversely, inflation-linked zero-coupon bonds have the highest duration of all inflation-linked bonds and do not have reinvestment risk. In addition, also the tax treatment affects the cash flow structure. In some bond markets, the inflation adjustment on the principal is treated as current income for tax purpose, while in other markets it is not. 

Z-spread The Z-spread or zero volatility spread calculates the yield spread of a corporate bond by taking a zero-coupon bond curve as benchmark. Conversely to other yield spreads, the Z-spread is constant. In fact, it is found as an iterative procedure, which is the yield spread required to get the equivalence between market price and the present value of all its cash flows. The Z-spread is given by Equation (8.2) ... 

Many Euro government bonds can be stripped, breaking them down into each of the single payments that they involve, that is, one flow for each remaining coupon payment and another one for the principal. With this procedure an n-year maturity coupon-bearing bond is transformed into n + 1 strips (zero coupon bonds), which can be traded separately in the market. Yet this market is much less liquid in the Eurozone than in the United States. 

In 1995 the SNDO launched the second inflation-linked bond, another zero but with a shorter maturity of 10 years (No. 3002, 0% 2004). At this time the SNDO decided to replace the common price anc-tions with multiple price auctions. Moreover, the SNDO opened a noncompetitive facility for small volumes in the auctions, so that small investors could enter the market. In February 1996 the SNDO launched two new bonds a 5-year zero-coupon bond (3003, 0% 2001) and a 12-year coupon bond (3101, 4% 2008). In June 1996 the 24-year coupon bond (3102, 4% 2020) was launched. The market continued to grow rapidly in 1997 and 1998. 

The futnre market price of a zero-coupon bond in this framework can be found by defining the reversion rate, P, the volatility, and the... 

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)... 

All bonds except zero-coupon bonds accrue interest on a daily basis that is then paid out on the coupon date. As mentioned earlier, the formulas discussed so far calculate bonds prices as of a coupon payment date, so that no accrued interest is incorporated in the price. In all major bond markets, the convention is to quote this so-called clean price. 

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. 

The bond and the money market are both risk-free and have identical payouts at time T, and neither will generate any cash flow between now and time T. Since the interest rates involved are constant, the bond must have a value equal to the initial investment in the money market account g -r(r q other words, equation (3.13) must hold. This is a restriction placed on the zero-coupon bond price by the requirement for markets to be arbitrage-free. 

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). 

Expression (3.14) is the formula for pricing zero-coupon bonds when the spot rate is the nonconstant instantaneous risk-free rate r(s) described above. This is the rate used in formulas (3.12), for valuing a money market account, and (3.15), for pricing a risk-free zero-coupon... 

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). 

Equilibrium interest rate models also exist. These make the same assumptions about the dynamics of the short rate as arbitrage models do, but they are not designed to match the current term structure. The prices of zero-coupon bonds derived using such models, therefore, do not match prices seen in the market. This means that the prices of bonds and interest rate derivatives are not given purely by the short-rate process. In brief, arbitrage models take as a given the current yield curve described by the... 

Using this expected price at period 1 and a discount rate of 5 percent (the six-month rate at point 0), the bonds present value at period 0 is 97.4399/(1 + 0.05/2), or 95.06332. As shown above, however, the market price is 95.0423. This demonstrates a very important principle in financial economics markets do not price derivative instruments based on their expected future value. At period 0, the one-year zero-coupon bond is a... 

The yield analysis described above considers coupon bonds as packages of zeros. How does one compare the yields of zero-coupon and coupon bonds A two-year zero is clearly the point of comparison for a coupon bond whose duration is two years. What about very long-dated zero-coupon bonds, though, for which no equivalent coupon Treasury is usually available The solution lies in the technique of stripping coupon Treasuries, which allows implied zero-coupon rates to be calculated, which can be compared with actual strip-market yields. 

This section describes the relationships among spot interest rates and the actual market yields on zero-coupon and coupon bonds. It explains how an implied spot-rate curve can be derived from the redemption yields and prices observed on coupon bonds, and discusses how this curve may be used to compare bond yields. Note that, in contrast with the common practice, spot rates here refer only to rates derived from coupon-bond prices and are distinguished from zero-coupon rates, which denote rates actually observed on zero-coupon bonds trading in the market. 

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. 

The potential profit from stripping a Treasury coupon depends on current market Treasury yields and the implied spot yield curve. Consider a hypothetical 5-year, 8 percent Treasury trading at par—and therefore offering a yield to maturity of 8 percent—in the yield curve environment shown in figure 16.2. A market maker buys the Treasury and strips it with the intention of selling the resulting zero-coupon bonds at the yields indicated in figure 16.2. 

Using this expected price at period 1 and a discount rate of 5 percent (the six-month rate at point 0), the bond s present value at period 0 is 97.4399/(1 -F 0.05/2), or 95-06332. As shown previously, however, the market price is 95.0423. This demonstrates a very important principle in financial economics markets do not price derivative instruments based on their expected future value. At period 0, the one-year zero-coupon bond is a riskier investment than the shorter-dated six-month zero-coupon bond. The reason it is risky is the uncertainty about the bond s value in the last six months of its life, which will be either 97-32 or 97.55, depending on the direction of six-month rates between periods 0 and 1. Investors prefer certainty. That is why the period 0 present value associated with the single estimated period 1 price of 97-4399 is higher than the one-year bond s actual price at point 0. The difference between the two figures is the risk premium that the market places on the bond. 

The term structure of interest rates is the spot rate yield curve spot rates are viewed as identical to zero-coupon bond interest rates where there is a market of liquid zero-coupon bonds along regular maturity points. As such a market does not exist anywhere the spot rate yield curve is considered a theoretical construct, which is most closely equated by the zero-coupon term structure derived from the prices of default-free liquid government bonds. 

Zero-coupon convertible bonds are well established in the market. When they are issued at a discount to par, they exhibit an implicit yield and trade essentially as coupon convertibles. Similarly, if they are issued at par but redeemed at a stated price above par, an implicit coupon is paid and so again these bonds trade in similar fashion to coupon convertibles. A zero-coupon bond issued at par and redeemed at par is a slightly different instrument for investors to consider. With these products, the buyer is making more of an equity play than with conventional convertibles, but with an element of capital protection retained. 

---