Zero-coupon curve

The zero-coupon curve is used in the asset-swap analysis, in which the curve is derived from the swap curve. Then, the asset-swap spread is the spread that allows us to receive the equivalence between the present value of cash flows and the current market price of the bond. 

The change in the short rate will result in a 50-basis point decline in all the expected future interest rates. However, this will not result in a uniform fall in all bond yields. The impact on the zero-coupon curve and the forward rate curve is shown in Figure 7.4. 

In chapter 2 of the companion volume to this book in the boxed-set library, Corporate Bonds and Structured Financial Products, we introduced the concept of the yield curve, and reviewed some preliminary issues concerning both the shape of the curve and to what extent the curve could be used to infer the shape and level of the yield curve in the future. We do not know what interest rates will be in the future, but given a set of zero-coupon (spot) rates today we can estimate the future level of forward rates using a yield curve model. In many cases however we do not have a zero-coupon curve to begin with, so it then becomes necessary to derive the spot yield curve from the yields of coupon bonds, which one can observe readily in the market. If a market only trades short-dated debt instruments, then it will be possible to construct a short-dated spot curve. 

This chapter considers some of the techniques used to fit the model-derived term structure to the observed one. The Vasicek, Brennan-Schwartz, Cox-Ingersoll-Ross, and other models discussed in chapter 4 made various assumptions about the nature of the stochastic process that drives interest rates in defining the term structure. The zero-coupon curves derived by those models differ from those constructed from observed market rates or the spot rates implied by market yields. In general, market yield curves have more-variable shapes than those derived by term-structure models. The interest rate models described in chapter 4 must thus be calibrated to market yield curves. This is done in two ways either the model is calibrated to market instruments, such as money market products and interest rate swaps, which are used to construct a yield curve, or it is calibrated to a curve constructed from market-instrument rates. The latter approach may be implemented through a number of non-parametric methods. 

It is not surprising that the net present value is zero. The zero-coupon curve is used to derive the discount factors that are then used to derive the forward rates that are used to determine the swap rate. As with any financial instrument, the fair value is its break-even price or hedge cost. The bank that is pricing this swap could hedge it with a series of FRAs transacted at the forward rates shown. This method is used to price any interest rate swap, even exotic ones. 

The first method equates a strips value with its spread to a bond having the same maturity. The main drawback of this rough-and-ready approach is that it compares two instruments with different risk profiles. This is particularly true for longer maturities. The second method, which aligns strip and coupon-bond yields on the basis of modified duration, is more accurate. The most common approach, however, is the third. This requires constructing a theoretical zero-coupon curve in the manner described above in connection with the relationship between coupon and zero-coupon yields. 

When the bond yield curve is flat, the spot curve is too. When the yield curve is inverted, the theoretical zero-coupon curve must lie below it. This is because the rates discounting coupon bonds earlier cash flows are higher than the rate discounting their final payments at redemption. In addition, the spread between zero-coupon and bond yields should decrease with maturity. 

When the yield curve is positive, the theoretical zero-coupon curve lies above the coupon curve. Moreover, the steeper the coupon curve, the steeper the zero-coupon curve. 

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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