Yields redemption

The continuously compounded gross redemption yield at time ton a default-free zero-coupon bond that pays 1 at maturity date 7 is x. We assume that the movement in X is described by... 

Gilt redemption yield Nelson Siegel curve... 

FIGURE 5.1 A Nelson and Siegel fitted yield curve and gilt redemption yield curve. 

The fitted curve is a close approximation to the redemption yield curve, and is also very smooth. However, the fit is inaccurate at the very short end, indicating an underpriced 6-month bond, and also does not approximate the long end of the curve. For this reason, B-spline methods are more commonly used. 

Term to Maturity Gilt Redemption Yield % Term to Maturity Gilt Redemption Yield %... 

The original form of this call pricing mechanism was the Spens clause, which required the issuer to call in the bonds at an above-market price if certain events detrimental to the interests of the bondholders took place. Following a decision by the International Primary Markets Association (IPMA) in July 2001, the market convention is that the Spens clause is no longer used, although it remains in many ontstanding issues. Instead, the market now uses a formula to calculate the redemption yield of a bond in the event it is called prior to maturity. This is set out in the UK Debt Management Office (DMO) paper dated 8 June 1998. 

Applying the indexation lag, this allows us to calculate estimated nominal values of all future cash flows which, knowing the current dirty nominal price, P, allows us to solve for an internal (nominal) rate of return—a nominal, semi-annual, gross redemption yield, y. Having applied our 3% inflation assumption, /, to get nominal future payments, we now remove it from the nominal yield using the simple formula... 

After the close of business each day the DMO publishes reference prices and the equivalent gross redemption yields for each gilt on its news screens. The final reference price is based on closing two-way prices supplied by each GEMM at the end of the day. The prices, previously referred to as CGO reference prices but now following the merger of the CGO with CREST, called DMO or gilt reference prices, are frequently used in the calculation of settlement proceeds in repo and stock loan transactions. 

In the past bond analysis was frequently limited to calculating gross redemption yield, or yield to maturity. Today basic bond math involves different concepts and calculations. These are described in several of the references for chapter 3, such as Ingersoll (1987), Shiller (1990), Neftci (1996), Jarrow (1996), Van Deventer (1997), and Sundaresan (1997). This chapter reviews the basic elements. Bond pricing, together with the academic approach to it and a review of the term structure of interest rates, are discussed in depth in chapter 3. 

The discussion so far has involved calculating the price of a bond given its yield. This procedure can be reversed to find a bond s yield when its price is known. This is equivalent to calculating the bond s internal rate of return, or IRR, also known as its yield to maturity or gross redemption yield... 

There are two solutions, only one of which gives a positive redemption yield. The positive solution is... 

Pluming in the appropriate values gives a linear approximation for the redemption yield of rm = 4.549 percent, which is near the solution obtained by solving the quadratic equation. 

Calculating the redemption yield of bonds that pay semiannual coupons involves the semiannual discounting of those payments. This approach is appropriate for most U.S. bonds and U.K. gilts. Government bonds in most of continental Europe and most Eurobonds, however, pay annual coupon payments. The appropriate method of calculating their redemption yields is to use annual discounting. The two yield measures are not directly comparable. 

Consider a bond with a dirty price—including the accrued interest the seller is entitled to receive—of 97.89, a coupon of 6 percent, and five years to maturity. FIGURE 1.6 shows the gross redemption yields this... 

These figures demonstrate the impact that the coupon-payment and discounting frequencies have on a bonds redemption yield calculation. Specifically, increasing the frequency of discounting lowers the calculated yield, while increasing the frequency of payments raises it. When comparing yields for bonds that trade in markets with different conventions, it is important to convert all the yields to the same calculation basis. 

The market convention is sometimes simply to double the semiannual yield to obtain the annualized yields, despite the fact that this produces an inaccurate result. It is only acceptable to do this for rough calculations. An annualized yield obtained in this manner is known as a hand equivalent yield. It was noted earlier that the one disadvantage of the YTM measure is that its calculation incorporates the unrealistic assumption that each coupon payment, as it becomes due, is reinvested at the rate rm. Another disadvantage is that it does not deal with the situation in which investors do not hold their bonds to maturity. In these cases, the redemption yield will not be as great. Investors might therefore be interested in other measures of return, such as the equivalent zero-coupon yield, considered a true yield. 

To review, the redemption yield measure assumes that... 

The second-order differential of the bond price equation with respect to the redemption yield r is... 

The asset swap spread is equal to the underlying asset s redemption yield spread over the government benchmark, minus the spread on the associated interest rate swap. The latter, which reflects the cost of convert-... 

The market quotes bonds with embedded options in terms of yield spreads. A cheap bond trades at a high spread, a dear one at a low spread. The usual convention is to quote the spread between the redemption yield of the bond being analyzed and that of a government bond having an equivalent maturity. This is not an accurate measure of the actual difference in value between the two bonds, however. The reason is that, as explained in chapter 1, the redemption yield computation unrealistically discounts all a bond s cash flows at a single rate. 

C, = the bond cash flow at time t P = the bonds fair price C = the annual coupon payment rm = the redemption yield n = the number of years to maturity... 

The equations for money yield and real yield can be interpreted as indicating what redemption yield to employ as the discount rate in calculating the present value of an index bond s future cash flows. From this perspective, equation (12.9) shows that the money yield is the appropriate rate for discounting money, or nominal, cash flows. Equation (12.12) shows that the real yield is the appropriate rate for discounting real cash flows. 

As an illustration, say the August 1999 redemption yield on the 5 percent Treasury maturing in 2009 was 5.17 percent and the money yield on the... 

One approach is to determine the expected inflation rate using the difference between the yield on a conventional and that on an indexed bond having the same maturity date, if such bonds exist, ignoring any lag effects. This, however, is a flawed measure of inflation, because the calculation of the indexed bond s redemption yield already assumes an expected inflation rate. 

Because the future values for the reference index are not known, it is not possible to calculate the redemption yield of an FRN. On the coupon-reset dates, the note will be priced precisely at par. Between these dates, it will trade very close to par, because of the way the coupon resets. If market rates rise between reset dates, the note will trade slightly below par if rates fall, it will trade slightly above par. This makes FRNs behavior very similar to that of money market instruments traded on a yield basis, although, of course, the notes have much longer maturities. FRNs can thus be viewed either as money market instruments or as alternatives to conventional bonds. Similarly, they can be analyzed using two approaches. 

Cash flow yield calculated in this way is essentially a redemption yield calculated assuming a prepayment rate to project the cash flows. As such, it has the same drawbacks as the redemption yield for a plain vanilla bond it assumes that all the cash flows will be reinvested at the same interest rate and that the bond will be held to maturity. In fact, the potential inaccuracy is even greater for a mortgage-backed bond because the frequency of interest payments is higher, which makes the reinvestment risk greater. The final yield of a mortgage-backed bond depends on the performance of the mortgages in the pool—specifically, their prepayment pattern. 

Given the nature of a mortgage-backed bond s cash flows, its exact yield cannot be calculated. Market participants, however, commonly compare an MBS s cash flow yield to the redemption yield of a government bond with a similar duration or a term to maturity similar to the MBS s average life. The usual convention is to quote the spread over the government bond. 

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... 

Spot yields cannot be directly observed in the market. They can, however, be computed from the observed prices of zero-coupon bonds, or strips, if a liquid market exists in these securities. An implied spot yield curve can also, as the previous section showed, be derived from coupon bonds prices and redemption yields. This section explores how the implied and actual strip yields relate to each other. 

FIGURE 16.10 shows the cash flow for the Treasury s principal strip. Its yield is 4.0751 percent, corresponding to a price of 67.10027 per 100 nominal, which represents a spread above the gross redemption yield of the coupon Treasury. This relationship is expected, given a positive yield... 

To differentiate redemption yield from other yield and interest rate measures described in this book, it will be referred to as rm. Note that this section is concerned with the gross redemption yield, the yield that results from payment of coupons without deduction of any withholding tax. The net redemption yield is what will be received if the bond is traded in a market where bonds pay coupon net, without withholding tax. It is obtained by multiplying the coupon rate Cby (1 — marginal tax rate). The net redemption yield is always lower than the gross redemption yield. 

The key assumption behind the YTM calculation has already been discussed—that the redemption yield rm remains stable for the entire life of the bond, so that all coupons are reinvested at this same rate. The assumption is unrealistic, however. It can be predicted with virtual certainty that the interest rates paid by instruments with maturities equal to those of the bond at each coupon date will differ from rm at some point, at least, during the life of the bond. In practice, however, investors require a rate of return that is equivalent to the price that they are paying for a bond, and the redemption yield is as good a measurement as any. 

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