Zero-swap rate

The short end of the swap curve, out to three months, is based on the overnight, 1-month, 2-month, and 3-month deposit rates. The short-end deposit rates are inherently zero-coupon rates and need only be converted to the base currency swap rate compounding frequency and day count convention. The following equation is solved to compute the continuously compounded zero-swap rate (r ) ... 

There is still a consistency problem if we want to price interest rate derivatives on zero bonds, like caplets or floorlets, and on swaps, like swaptions, at the same time within one model. The popular market models concentrate either on the valuation of caps and floors or on swaptions, respectively. Musiela and Rutkowski (2005) put it this way We conclude that lognormal market models of forward LIBORs and forward swap rates are inherently inconsistent with each other. A challenging practical question of the choice of a benchmark model for simultaneous pricing and hedging of LIBOR and swap derivatives thus arises. ... 

As shown in Eigures 1.4 and 1.5, with this swap structuring, the asset-swap spread for HERIM is 39.5 bp and for TKAAV is 39.1 bp. These represent the spreads that will be received if each bond is purchased as an asset-swap package. In other words, the ASW spread provides a measure of the difference between the market price of the bond and the value of the cash flows evaluated using zero-coupon rates. 

In selecting the model, a practitioner will select the market variables that are incorporated in the model these can be directly observed such as zero-coupon rates or forward rates, or swap rates, or they can be indeterminate such as the mean of the short rate. The practitioner will then decide the dynamics of these market or state variables, so, for example, the short rate may be assumed to be mean reverting. Finally, the model must be calibrated to market prices so, the model parameter values input must be those that produce market prices as accurately as possible. There are a number of ways that parameters can be estimated the most common techniques of calibrating time series data such as interest rate data are general method of moments and the maximum likelihood method. For information on these estimation methods, refer to the bibliography. 

At the inception of the swap, the terms of the swap will be such that the present value of the floating-rate payments is equal to the present value of the fixed-rate payments. That is, the value of the swap is equal to zero at its inception. This is the fundamental principle in determining the swap rate (i.e., the fixed rate that the fixed-rate payer will make). 

The fixed-rate payer will require that the present value of the fixed-rate payments that must be made based on the swap rate not exceed the 14,052,917 payments to be received from the floating-rate payments. The fixed-rate receiver will require that the present value of the fixed-rate payments to be received is at least as great as the 14,052,917 that must be paid. This means that both parties will require a present value for the fixed-rate payments to be 14,052,917. If that is the case, the present value of the fixed-rate payments is equal to the present value of the floating-rate payments and therefore the value of the swap is zero for both parties at the inception of the swap. The interest rates that should be used to compute the present value of the fixed-rate payments are the same interest rates as those used to discount the floating-rate payments. 

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 ... 

We consider three-or-six-month EURIBOR swap yields with maturities ranging from one year to 10 years and find recursively equivalent zero-coupon rates. Swap yields are par yields so the zero-coupon rate with maturity two years R(0,2) is obtained as the solution to the following equation ... 

As discussed above, vanilla swap rates are often quoted as a spread that is a function mainly of the credit spread required by the market over the risk-free government rate. This convention is logical, because government bonds are the principal instrument banks use to hedge their swap books. It is unwieldy, however, when applied to nonstandard tailor-made swaps, each of which has particular characteristics that call for particular spread calculations. As a result, banks use zero-coupon pricing, a standard method that can be applied to all swaps. 

As explained in chapter 3, zero-coupon, or spot, rates are true interest rates for their particular terms to maturity. In zero-coupon swap pricing, a bank views every swap, even the most complex, as a series of cash flows. The zero-coupon rate for the term from the present to a cash flows payment date can be used to derive the present value of the cash flow. The sum of these present values is the value of the swap. 

Expression (7.9) formalizes the hootstrapping process described in chapter 3. Essentially, the -year discount factor is computed using the discount factors for years one to n- and the -year swap or zero-coupon rate. Given the discount factor for any period, that period s zero-coupon, or spot, rate can be derived using (7.9) rearranged as (7.10). 

Equations (7-8) and (7.l4) can be combined to obtain (7.16) and (7.17), the general expressions for, respectively, an n-period swap rate and an -period zero-coupon rate. 

As noted earlier,a newly transacted interest rate swap denotes calculating the swap rate that sets the net present value of the cash flows to zero. Valuation signifies the process of calculating the net present value of an existing swap by setting its fixed rate at the current market rate. Consider a plain vanilla interest rate swap with the following terms ... 

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. 

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. 

A more accurate approach m ht be the one used to price interest tate swaps to calculate the present values of future cash flows usit discount tates determined by the markets view on where interest rates will be at those points. These expected rates ate known as forward interest rates. Forward rates, however, are implied, and a YTM derived using them is as speculative as one calculated using the conventional formula. This is because the real market interest rate at any time is invariably different from the one implied earlier in the forward markets. So a YTM calculation made using forward rates would not equal the yield actually realized either. The zero-coupon rate, it will be demonstrated later, is the true interest tate for any term to maturity. Still, despite the limitations imposed by its underlying assumptions, the YTM is the main measure of return used in the markets. 

Put simply, the Z-spread is the basis point spread that would need to be added to the implied spot yield curve such that the discounted cash flows of the bond are equal to its present value (its current market price). Each bond cash flow is discounted by the relevant spot rate for its maturity term. How does this differ from the conventional asset-swap spread Essentially, in its use of zero-coupon rates when assigning a value to a bond. Each cash flow is discounted using its own particular zero-coupon rate. The bond s price at any time can be taken to be the market s value... 

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. 

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. 

The fundamental principle of valuation is that the value of any financial asset is equal to the present value of its expected future cash flows. This principle holds for any financial asset from zero-coupon bonds to interest rate swaps. Thus, the valuation of a financial asset involves the following three steps ... 

The reader may notice that the value of the example default swap is close to zero this is not coincidental and does warrant some explanation. It is often heard in the market that default swap spreads are representative of default probabilities—it is clear from our example that the hazard rate, X, equals 5.00%, but the premium of the default swap is only 4.00%. The reason for this discrepancy is not complex and resnlts directly from our assumption of the recovery value, R = 20%. 

The default swap market is not unlike the lottery ticket. What if the shipping and handling fee for the winning ticket was unknown or turned out to be zero In that case, if an investor observed these lottery tickets trading at a price of 4, it may appear that the probability of winning was simply 4.00%. In the case of a default swap this is what is referred to as the risk-neutral probability of default. The risk-neutral probability of our default swap is approximately equal to the premium of 4.00%. By applying the lottery ticket example to our default swap, it is easy to see how the hazard rate is dependent on both the risk-neutral probability as well as the recovery value assumption, and thus can be approximated hy X = P/(l - R). 

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. 

Our starting point is a set of zero curve tenors (or discount factors) obtained from a collection of market instruments such as cash deposits, futures, swaps, or coupon bonds. We therefore have a set of tenor points and their respective zero rates (or discount factors). The mathematics of cubic splines is straightforward, but we assume a basic understanding of calculus and a familiarity with solving simultaneous linear equations by substitution. An account of the methods analyzed in this section is given in Burden and Faires (1997), which has very accessible text on cubic spline interpolation. ... 

Using equation 14.16, we can build a forward inflation curve provided we have the values of the index at present, as well as a set of zero-coupon bond prices of required credit quality. Following standard yield curve analysis, we may build the term structure from forward rates and therefore imply the real yield curve, or alternatively we may construct the real curve and project the forward rates. However, if we are using inflation swaps for the market price inputs, the former method is preferred because IL swaps are usually quoted in terms of a forward index value. 

The zero-coupon curve is used in the asset swap valuation. This curve is derived from the swap curve, so it is the implied zero-coupon curve. The asset swap spread is the spread that equates the difference between the present value of the bond s cash flows, calculated using the swap zero rates, and the market price of the bond. This spread is a function of the bond s market price and yield, its cash flows, and the implied zero-coupon interest rates. ... 

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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