Zero-coupon Treasury yield curves

In this chapter, we perform a factor analysis of the zero-coupon euro interbank yield curve, and also of zero-coupon Treasury yield curves from five individual countries, France, Germany, Italy, Spain, and the Netherlands, so as to isolate the key aspects of the dynamics of term structures of interest rates in the Eurozone. 

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. 

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

FIGURE 5.2 is the graph of the discount function derived by bootstrapping from the U.S. Treasury prices as of December 23, 2003- FIGURE 5.3 shows the zero-coupon yield and forward-rate curves corresponding to this discount function. Compare these to the yield curve in FIGURE 5.1... 

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. 

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. 

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