Zero-coupon yield curve

Z-spread is an alternative spread measure to the ASW spread. This type of spread uses the zero-coupon yield curve to calculate the spread, in which in this case is assimilated to the interest-rate swap curve. Z-spread represents the spread needful in order to obtain the equivalence between the present value of the bond s cash flows and its current market price. However, conversely to the ASW spread, the Z-spread is a constant measme. 

From an elementary understanding of the markets, we know that there is a relationship between a set of discount factors, and the discount function, the par yield curve, the zero-coupon yield curve and the forward yield curve. If we know one of these functions, we may readily compute the other three. In practice, although the zero-coupon yield curve is directly observable from the yields of zero-coupon... 

As above, assuming a constant average inflation rate, which is then used to calculate the value of the bond s coupon and redemption payments. The duration of the cash flow is then calculated by observing the effect of a parallel shift in the zero-coupon yield curve. By assuming a constant inflation rate and constant increase in the cash flow stream, a further assumption is made that the parallel shift in the yield curve is as a result of changes in real yields, not because of changes in inflation expectations. Therefore, this duration measure becomes in effect a real yield duration ... 

The probabilities of each of these occurrences are 10%, 80% and 10%, respectively that is, the most likely scenario is a rise in the short rate from 6% to 8%. For each scenario, we assume that the short rate approaches the expected Iraig-term level in exponential fashion. The expected interest-rate scenario, therefore, is a rise from 6% to 8%. From Figure 7.2, we see that the forward rate curve behaves differently to expected future short-rate levels. The forward rates peak at around 12-14 years and then steadily decline as the term to maturity increases. The zero-coupon yield curve, which can be derived from the forward yield curve, has a different shape and starts to decline from the 20-year term period. 

Figure 7.2 suggests that the unbiased expectatirais hypothesis, which states that forward rates are equal to the expected level of future short-term rates, is incorrect, and so it is not valid to calculate par and zero-coupon yield curves using the expected short-rate curve. Instead, the forward rate curve... 

Expected short-rate curve Zero-coupon yield curve Forward rate... 

FIGURE 7.3 Zero-coupon yield curves calculated using expected short- and forward rates. 

It is important for a zero-coupon yield curve to be constructed as accurately as possible. This because the curve is used in the valuation of a wide range of instruments, not only conventional cash market coupon bonds, which we can value using the appropriate spot rate for each cash flow, but other interest-rate products such as swaps. 

We derive daily zero-coupon yield curves from five countries of the Eurozone (France, Germany, Italy, Spain, and the Netherlands) during the period from 2 January 2001 to 21 August 2002, using zero-coupon rates with 26 different maturities ranging from one month to 30 years. The yield curves are extracted from daily Treasury bond market prices by using a standard cubic B-splines method. Our input baskets are composed of... 

Least-squared methods used to derive the current interbank curve are very similar to those used to derive the current nondefault Treasury curve. After converting market data into equivalent zero-coupon rates, the zero-coupon yield curve is derived using a two-stage process, first writing zero-coupon rates as a B-spline function, and then fitting them through an ordinary least-squared method. 

We first consider the global fraction of the total variance of the zero-coupon yield curve changes that is accounted for by the five first factors (see Exhibit 24.1). 

FIGURE 3.1 U.S. Treasury Zero-Coupon Yield Curve in September 2000 ... 

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 ... 

Some of the newer models refer to parameters that are difficult to observe or measure direcdy. In practice, this limits their application much as B-S is limited. Usually the problem has to do with calibratii the model properly, which is crucial to implementing it. Galibration entails inputtii actual market data to create the parameters for calculating prices. A model for calculating the prices of options in the U.S. market, for example, would use U.S. dollar money market, futures, and swap rates to build the zero-coupon yield curve. Multifactor models in the mold of Heath-Jarrow-Morton employ the correlation coefficients between forward rates and the term structure to calculate the volatility inputs for their price calculations. 

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. 

The conventional approach for analyzing an asset swap uses the bonds yield-to-maturity (YTM) in calculating the spread. The assumptions implicit in the YTM calculation (see Chapter 2) make this spread problematic for relative analysis, so market practitioners use what is termed the Z-spread instead. The Z-spread uses the zero-coupon yield curve to calculate spread, so is a more realistic, and effective, spread to use. The zero-coupon curve used in the calculation is derived from the interest-rate swap curve. 

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