Duration and, to a lesser extent, convexity are core topics in the study of bond math, right up there with price and yield calculations and conversions, and yield curve analysis. These statistics are fundamentally mathematical and are derived using algebra and calculus. Duration is sometimes just a building block for more developed models of risk. For example, value-at- risk (VaR) analysis includes the effect of varying volatilities and correlations for points along the yield curve. Although VaR commonly is used for risk measurement in financial institutions and is subject to its own limitations and misapplications, it goes “beyond duration” in terms of mathematics and statistics. But to get there, you need a solid foundation in classic duration analysis.

In its primary application, duration estimates the change in market value corresponding to a change in the yield to maturity. This estimation is improved with the convexity adjustment. Most literature on this topic focuses on a single fixed-income bond. But if you owned only one bond, why would you need to estimate changes in market value? If you have a financial calculator and some good bond math formulas, or if you have an Excel spreadsheet and can open the financial functions, you can get the new bond price for any change in its yield to maturity.

In practice, we need to think about portfolios of bonds and the average duration and convexity statistics. Those averages will be helpful in risk management and in structuring the portfolio. How the market value of the overall portfolio, and specific holdings within it, move when the benchmark Treasury yield curve shifts becomes relevant. We need to think about the difference between classic Macaulay and modified yield duration and convexity, which are fine for individual securities, and curve duration and convexity, which are more applicable to measuring risk in portfolios of bonds.

In this chapter we worked with traditional fixed-income and zero-coupon bonds. Now we turn to other types of bonds, in particular, floating-rate and inflation-indexed bonds, and then to a very important type of derivative contract, interest rate swaps. We will have to use all of our bond math core topics to understand them.

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