Zero-coupon bonds duration

To obtain the price of an inflation-linked bond, it is necessary to determine the value of coupon payments and principal repayment. Inflation-linked bonds can be structured with a different cash flow indexation. As noted above, duration, tax treatment and reinvestment risk, are the main factors that affect the instrument design. For instance, index-aimuity bmids that give to the investor a fixed annuity payment and a variable element to compensate the inflation have the shortest duration and the highest reinvestment risk of aU inflation-linked bonds. Conversely, inflation-linked zero-coupon bonds have the highest duration of all inflation-linked bonds and do not have reinvestment risk. In addition, also the tax treatment affects the cash flow structure. In some bond markets, the inflation adjustment on the principal is treated as current income for tax purpose, while in other markets it is not. 

Chapter 4 provides a comprehensive discussion of duration. Duration is the change in the price of a bond as a result of a very small shift in its yield. In other words, duration measures the sensitivity of the price of a bond to changes in its yield. An increasingly common measure of duration is effective duration, which is the measure of price sensitivity due to a small parallel shift in the spot curve. One would immediately realise that these two definitions of duration would give identical results for zero-coupon bonds and different results for most other instruments. 

Duration varies with maturity, coupon, and yield. Broadly, it increases with maturity. A bonds duration is generally shorter than its maturity. This is because the cash flows received in the early years of the bond s life have the greatest present values and therefore are given the greatest weight. That shortens the avetj e time in which cash flows are received. A zero-coupon bond s cash flows are all received at redemption, so there is no present-value weighting. Therefore, a zero-coupon bond s duration is equal to its term to maturity. 

Say a trader holds a long position of 1 million of the 8 percent bond maturing in 2019. The bond s modified duration is 11.14692, and its price is 129.87596. Its basis point value is therefore 0.14477. The trader decides to protect the position against a rise in interest rates by hedging it using the zero-coupon bond maturing in 2009, vi/hich has a BPV of 0.05549. Assuming that the yield beta is 1, what nominal value of the zero-coupon bond must the trader sell ... 

Zero-coupon bonds don t pose these problems, because their durations are identical to their terms to maturity. This potentially increases their attractiveness as investments. A five-year zero-coupon bond has a duration of five years when purchased after two years, its duration is three years, no matter what interest rates have done. A long-dated zero-coupon bond can thus be safely used to match a long-dated liability. 

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. 

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 ... 

Portfolio convexity depends on the distribution of cash flows in the portfolio. A portfolio with an even distribution of cash flows has higher convexity than one where cash flows are concentrated in a particular maturity bucket, assuming equal duration and no optionality. By extension, considering a bond to be a portfolio of cash flows, the obvious conclusion is that bonds with higher coupons have higher convexity than bonds with low or zero coupons. 

There are five basic methods of linking the cash flows from a bond to an inflation index interest indexation, capital indexation, zero-coupon indexation, annuity indexation, and current pay. Which method is chosen depends on the requirements of the issuers and of the investors they wish to attract. The principal factors considered in making this choice, according to Deacon and Derry (1998), are duration, reinvestment risk, and tax treatment. 

The first method equates a strips value with its spread to a bond having the same maturity. The main drawback of this rough-and-ready approach is that it compares two instruments with different risk profiles. This is particularly true for longer maturities. The second method, which aligns strip and coupon-bond yields on the basis of modified duration, is more accurate. The most common approach, however, is the third. This requires constructing a theoretical zero-coupon curve in the manner described above in connection with the relationship between coupon and zero-coupon yields. 

Zero-coupon indexation. Zero-coupon indexed bonds have been issued in Sweden. As their name implies, they pay no coupons the entire inflation adjustment occurs at maturity, applied to their redemption value. These bonds have the longest duration of all indexed securities and no reinvestment risk. 

Part One, Introduction to Bonds, covers bond mathematics, including pricing and yield analytics. This includes modified duration and convexity. Chapters also cover the concept of spot (zero-coupon) and forward rates, and the rates implied by market bond prices and yields yield-curve fitting techniques an account of spline fitting using regression techniques and an introductory discussion of term structure models. 

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