Modified Duration, and Convexity

Most bonds pay a part of their total return during their lifetimes, in the form of coupon interest. Because of this, a bonds term to maturity does not reflect the true period over which its return is earned. Term to maturity also fails to give an accurate picture of the trading characteristics of a bond or to provide a basis for comparing it with other bonds having similar maturities. Clearly, a more accurate measure is needed. 

The average time until receipt of a bond s cash flows, weighted according to the present values of these cash flows, measured in years, is known as duration or Macaulays duration, referring to the man who introduced the concept in 1938—see Macaulay (1999) in References. Macaulay introduced duration as an alternative for the length of time remaining before a bond reached maturity. 

Duration is a measure of price sensitivity to interest rates—that is, how much a bond s price changes in response to a change in interest rates. In mathematics, change like this is often expressed in terms of differential equations. The price-yield formula for a plain vanilla bond, introduced in chapter 1, is repeated as (2.1) below. It assumes complete years to maturity, annual coupon payments, and no accrued interest at the calculation date. 

N = the number of years to maturity, and so the number of interest periods for a bond paying an annual coupon M = the maturity payment 

Chapter 1 showed that the price and yield of a bond are two sides of the same relationship. Because price T is a function of yield r, we can differentiate the price/yield equation at (2.1), as shown in (2.2). Taking the first derivative of this expression gives (2.2). 

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The first two chapters of this section discuss bond pricing and yields, moving on to an explanation of such traditional interest rate risk measures as modified duration and convexity. Chapter 3 looks at spot and forward rates, the derivation of such rates from market yields, and the yield curve. Yield-curve analysis and the modeling of the term structure of interest rates are among the most heavily researched areas of financial economics. The treatment here has been kept as concise as possible. The References section at the end of the book directs interested readers to accessible and readable resources that provide more detail. 

The modified duration and convexity methods we have described are only suitable for use in the analysis of conventional fixed-income instruments with known fixed cash flows and maturity dates. They are not satisfactory for use with bonds that contain embedded options such as callable bonds or instruments with unknown final redemption dates such as mortgage-backed bonds. For these and other bonds that exhibit uncertainties in their cash flow pattern and redemption date, so-called option-adjusted measures are used. The most common of these is option-adjusted spread (OAS) and option-adjusted duration (OAD). The techniques were developed to allow for the uncertain cash flow structure of non-vanilla fixed-income instruments, and model the effect of the option element of such bonds. 

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. 

If the yield rises to 7.50 percent, a change of 200 basis points, the convexity adjustment that would be made to the price change calculated using modified duration and equation (2.21) is... 

In principle, a bond with greater convexity should fall in price less when yields rise than a less-convex one, and rise in price more when yields fall. This is true because convexity is usually positive, so it lessens the price decline produced by rises in yield and increases the price rise produced by falls in yield. Thus, all else being equal, the higher the convexity of a bond the more desirable it should be to investors. The actual premium attached to higher convexity is a function of current yield levels and market volatility. Remember that modified duration and... 

The convexity measure increases with the square of maturity it decreases as both coupon and yield rise. It is a function of modified duration, so index-linked bonds, which have greater duration than conventional bonds of similar maturities, also have greater convexity. For a conventional vanilla bond, convexity is almost always positive. Negative convexity occurs most frequently with callable bonds. 

As with duration, there is little difference between modified convexity and effective convexity for option-free bonds. However, for bonds with embedded options there can be quite a difference between the calculated modified convexity and effective convexity measures. In fact, for all option-free bonds, either convexity measure will have a positive value. For bonds with embedded options, the calculated effective convexity measure can be negative when the calculated modified convexity measure is positive. 

Duration is a first-order measure of interest rate risk, using first-order derivatives. If the relationship between price and yield is plotted on a graph, it forms a curve. Duration indicates the slope of the tangent at any point on this curve. A tangent, however, is a line and, as such, is only an approximation of the actual curve—an approximation that becomes less accurate the farther the bond yield moves from the original point. The magnitude of the error, moreover, depends on the curvature, or convexity, of the curve. This is a serious drawback, and one that applies to modified as well as to Macaulay duration. 

Change of the process duration on the sample surface from periphery to center Relative density of active and inert liquid, p Sequence of active/inert liquids delivery Change of the refraetive index in modified polymer layer fi om periphery to center of the sample Regime of active/inert liquids delivery (sign minus corresponds to convex lens samples, sign plus - to concave ones)... 

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