Zero-coupon bonds calculating yields

Therefore, once we have a full description of the random behaviour of the short-rate r, we can calculate the price and yield of any zero-coupon bond at any time, by calculating this expected value. The implication is clear specifying the process r t) determines the behaviour of the entire term structure so, if we wish to build a term structure model, we need only (under these assumptions) specify the process for r t). 

In order to calculate the range of implied forward rates, we require the term stmcture of spot rates for all periods along the continuous discount function. This is not possible in practice, because a bond market will only contain a finite number of coupon-bearing bonds maturing on discrete dates. While the coupon yield curve can be observed, we are then required to fit the observed curve to a continuous term structure. Note that in the United Kingdom gilt market, for example there is a zero-coupon bond market, so that it is possible to observe spot rates directly, but for reasons of liquidity, analysts prefer to use a fitted yield curve (the theoretical curve) and compare this to the observed curve. 

Z-spread The Z-spread or zero volatility spread calculates the yield spread of a corporate bond by taking a zero-coupon bond curve as benchmark. Conversely to other yield spreads, the Z-spread is constant. In fact, it is found as an iterative procedure, which is the yield spread required to get the equivalence between market price and the present value of all its cash flows. The Z-spread is given by Equation (8.2) ... 

Note that, even though these bonds pay no actual coupons, their prices and yields must be calculated on the basis of quasi-coupon periods, which are based on the interest periods of bonds denominated in the same currency. A U.S. dollar or a sterling five-year zero-coupon bond, for example, would be assumed to cover ten quasi-coupon peri-... 

B. Calculate the price of a French government zero-coupon bond with precisely five years to maturity, with the same required yield of 5.40 percent. Note that French government bonds pay coupon annually. 

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. 

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. 

As above, assuming a constant average inflation rate, which is then used to calculate the value of the bond s coupon and redemption payments. The duration of the cash flow is then calculated by observing the effect of a parallel shift in the zero-coupon yield curve. By assuming a constant inflation rate and constant increase in the cash flow stream, a further assumption is made that the parallel shift in the yield curve is as a result of changes in real yields, not because of changes in inflation expectations. Therefore, this duration measure becomes in effect a real yield duration ... 

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. 

Once the CPIs have heen forecast, the bond s yield can be calculated. Assuming that the analysis is carried out on a coupon date so that accrued interest is zero, the money yield of a bond paying semiannual coupons is calculated by solving equation (12.9) for ri. 

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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