Zero-coupon rates

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. 

A landmark development in interest-rate modelling has been the specification of the dynamics of the complete term stracture. In this case, the volatility of the term structure is given by a specified functiOTi, which may be a function of time, term to maturity or zero-coupon rates. A simple approach is described in the Ho-Lee model, in which the volatility of the term structure is a parallel shift in the yield curve, the extent of which is independent of the current time and the level of current interest rates. The Ho-Lee model is not widely used, although it was the basis for the HJM model, which is widely used. The HJM model describes a process whereby the whole yield curve evolves simultaneously, in accordance with a set of volatility term structures. The model is usually described as being one that describes the evolution of the forward rate however, it can also be expressed in terms of the spot rate or of bond prices (see, e.g., James and Webber (1997), Chapter 8). For a more detailed description of the HJM framework refer to Baxter and Rennie (1996), Hull (1997), Rebonato (1998), Bjork (1996) and James and Webber (1997). Baxter and Reimie is very accessible, while Neftci (1996) is an excellent introduction to the mathematical background. 

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. 

We assume that the zero-coupon rate r-term structure is flat until the time s and that forward rates are flat at /. The value of 1 to be received at time t>s is given by ... 

This can be illustrated with an example. Consider a situation where the zero-coupon rates term structure is flat at 6% for 30 years and that forward rates are flat at/for terms from 30 to 100 years. This results in the price of a 30-year bond with a coupon of 6% and a redemption value of 100 having a price of par, shown below ... 

Dybvig, P., Ingersoll, T., Ross, S., 1996. Long forward and zero-coupon rates can never fall. J. Bus. 69 (1), 1-25. 

When valuing an option written on say, an equity the price of the underlying asset is the current price of the equity. When pricing an interest-rate option the underlying is obtained via a random process that described the instantaneous risk-free zero-coupon rate, which is generally termed the short 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 ) ... 

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... 

We consider EURIBOR rates with maturities ranging from one day to one year. These rates, expressed on an actual/360 basis, are first converted into equivalent zero-coupon rates on an Actual/365 basis. Eor example, on 1 January 1999, the 1-month EURIBOR rate was equal to 2.5%. Using the Actual/365 basis, the equivalent zero-coupon rate (denoted by R(0,l/12)) is given by... 

We consider futures 3-month EURIBOR futures contracts and find zero-coupon rates from raw data. The price of a 3-month LIBOR contract is given by 100 minus the underlying 3-month forward rate. For example, on 15 March 1999, the 3-month LIBOR rate was 3%, and... 

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 ... 

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. 

EXHIBIT 24.7 Sensitivity of Zero-Coupon Rate Changes with Respect to Factor 1... 

For the period as a whole, the first factor (see Exhibit 24.7) may actually be regarded as a level factor since it affects similarly all zero-coupon rates, except for the portion (1 month-1 year), which moves differently. Displaying the sensitivity of interest rates with respect to the second factor, Exhibit 24.8 shows a decreasing shape, first positive for short-term maturities then negative beyond. Hence, the second factor may be regarded as a rotation factor around a medium maturity between two and four years depending on the country we consider. The third factor (see Exhibit 24.9) has different effects on intermediate maturities as opposed to extreme maturities (short and long). Hence, it may be interpreted as a curvature factor. 

Some texts refer to the graph of coupon-bond yields plotted against maturities as the term structure of interest rates. It is generally accepted, however, that this phrase should be used for zero-coupon rates only and that the graph of coupon-bond yields should be referred to instead as the yield curve. Of course, given the law of one price—which holds that two bonds having the same cash flows should have the same values—the zero-coupon term structure is related to the yield to maturity curve and can be derived from it. 

The first term-structure models described in the academic literature explain interest rate behavior in terms of the dynamics of the short rate. This term refers to the interest rate for a period that is infinitesimally small. (Note that spot rate and zero-coupon rate are terms used often to... 

McCulloch (1971) proposes a more practical approach, using polynomial splines. This method produces a fimction that is both continuous and linear, so the ordinary least squares regression technique can be employed. A 1981 study by James Langetieg and Wilson Smoot, cited in Vasicek and Fong (1982), describes an extended McCulloch method that fits cubic splines to zero-coupon rates instead of the discount fimction and uses nonlinear methods of estimation. 

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. 

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 ... 

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. 

Equation (7.17) states that the zero-coupon rate is the geometric average of one plus the forward rates. The w-period forward rate is obtained using the discount factors for periods n and n-. The discount factor for the complete period is obtained by multiplying the individual discount factors together. Exactly the same result would be obtained using the zero-coupon interest rate for the whole period to derive the discount factor. ... 

PERIOD ZERO-COUPON RATE % 1 5.5 DISCOUNT FACTOR 0.947867298 FORWARD RATE % 5.5... 

PERIOD ZERO-COUPON RATE % DISCOUNT FACTOR FORWARD RATE %... 

Zero-coupon and forward rates are also related in another way. If the zero-coupon rate rs and the forward rate r are transformed to their continuously compounded equivalent rates, ln(l + rs and ln(l + rfi), the result is the following expression, which derives the continuously compounded zero-coupon rate as the simple average of the continuously compounded forward rates ... 

The yield analysis described above considers coupon bonds as packages of zeros. How does one compare the yields of zero-coupon and coupon bonds A two-year zero is clearly the point of comparison for a coupon bond whose duration is two years. What about very long-dated zero-coupon bonds, though, for which no equivalent coupon Treasury is usually available The solution lies in the technique of stripping coupon Treasuries, which allows implied zero-coupon rates to be calculated, which can be compared with actual strip-market yields. 

This section describes the relationships among spot interest rates and the actual market yields on zero-coupon and coupon bonds. It explains how an implied spot-rate curve can be derived from the redemption yields and prices observed on coupon bonds, and discusses how this curve may be used to compare bond yields. Note that, in contrast with the common practice, spot rates here refer only to rates derived from coupon-bond prices and are distinguished from zero-coupon rates, which denote rates actually observed on zero-coupon bonds trading in the market. 

Equation (16.7) 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... 

Despite their supply-and-demand-induced divergence from zero-coupon rates, implied spot rates are important because they enable inves-... 

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. 

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