Swap curves derivation

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. 

Another approach is to compare the floating-rate note with a derived yield of a fixed-rate bond by using an interest rate swap curve matched with floater coupons. Figure 10.4 shows the Bloomberg YASN screen for Mediobanca float... 

The swap curve depicts the relationship between the term structure and swap rates. The swap curve consists of observed market interest rates, derived from market instruments that represent the most liquid and dominant instruments for their respective time horizons, bootstrapped and combined using an interpolation algorithm. This section describes a complete methodology for the construction of the swap term structure. 

In deriving the swap curve, the inputs should cover the complete term structure (i.e., short-, middle-, and long-term parts). The inputs should be observable, liquid, and with similar credit properties. Using an interpolation methodology, the inputs should form a complete, consistent, and smooth yield curve that closely tracks observed market data. Once the complete swap term structure is derived, an instrument is marked to market by extracting the appropriate rates off the derived curve. 

The technique for constructing the swap term structure, as constructed by market participants for marking to market purposes, divides the curve into three term buckets. The short end of the swap term structure is derived using interbank deposit rates. The middle area of the swap curve is derived from either forward rate agreements (FRAs) or interest rate futures contracts. The latter requires a convexity adjustment to render it equivalent to FRAs. The long end of the term structure is constructed using swap par rates derived from the swap market. 

To derive the swap term structure, observed market interest rates combined with interpolation techniques are used also, dates are constructed using the applicable business-day convention. Swaps are frequently con-strncted nsing the modified following bnsiness-day convention, where the cash flow occurs on the next business day unless that day falls in a different month. In that case, the cash flow occurs on the immediately preceding business day to keep payment dates in the same month. The swap curve yield calculation convention frequently differs by currency. Exhibit 20.2 lists the different payment frequencies, compounding frequencies, and day count conventions, as applicable to each currency-specific interest rate type. 

The long end of the swap curve is derived directly from observable coupon swap rates. These are generic plain vanilla interest rate swaps with fixed rates exchanged for floating interest rates. The fixed swap rates are quoted as par rates and are usually compounded semiannually (see Exhibit 20.2). The bootstrap method is used to derive zero-coupon interest rates from the swap par rates. Starting from the first swap rate, given all the continuously compounded zero rates for the coupon cash flows prior to maturity, the continuously compounded zero rate for the term of the swap is bootstrapped as follows ... 

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. 

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. 

The two previous chapters introduced and described a fractiOTi of the most important research into interest-rate models that has been carried out since the first model, presented by Oldrich Vasicek, appeared in 1977. These models can be used to price derivative seciuities, and equitibrium models can be used to assess fair value in the bond market. Before this can take place however, a model must be fitted to the yield curve, or calibrated In practice, this is carried out in two ways the most popular approach involves calibrating the model against market interest rates given by instruments such as cash Libor deposits, futures, swaps and bonds. The alternative method is to model the yield curve from the market rates and then calibrate the model to this fitted yield curve. The first approach is common when using, for example extended Vasicek... 

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. 

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