Duration and Convexity

Duration and convexity are statistics that estimate the sensitivity of the market value of an asset or liability to a change in interest rates. Usually the asset or liability is a fixed-income bond, but as measures of rate sensitivity, they apply to all sorts of securities and derivatives. We can ask meaningfully about the duration and convexity of a floating-rate note, an inflation-indexed bond, or an interest rate swap. That discussion will have to wait until Chapters 7 and 8. This chapter focuses on the risk statistics applicable to a typical fixed-rate or zero-coupon bond.

We start with classic yield duration – the sensitivity of the bond price to a change in its yield to maturity. This leads to the well-known Macaulay and modified duration statistics. Yield convexity is the second-order effect of that yield change. The beauty of yield duration and convexity is that they are based on fundamental mathematical properties of the bond. That means closed-form formulas can be derived for the statistics using algebra and calculus. Then we move on to other descriptions of change in interest rates – the sensitivity of the bond price to a shift in the benchmark Treasury yield curve. I call these curve duration and curve convexity.

Before diving into the bond math, let's get a sense of interest rate sensitivity using an admittedly contrived scenario. Suppose that you are the fixed-income strategist for an aggressively managed, high-yield, international bond fund. You believe that the market prices of some country's long-term bonds will rally in the next week as the market digests what you expect to be very positive news about economic conditions. In particular, two long-term sovereign bonds are trading at deeply discounted prices to yield 20%. Both bonds have an annual coupon rate of 6%, paid once a year. One bond matures in 20 years and the other in 30 years; otherwise they are identical. Which bond do you recommend, assuming that you anticipate both yields to drop by 100 basis points from 20% to 19% – the 20-year or the 30-year bond?

I've posed this problem over the last 25 years to hundreds of students, including some emerging-market, fixed-income traders. Virtually all recommend the 30-year bond; you probably do, too. Your thinking probably is that, other things being equal (meaning the coupon rate, yield to maturity, payment frequency, default risk, liquidity, taxation), the longer-term 30-year bond gains or loses more value on a percentage basis than the shorter-term 20-year bond given the same change in yield. That's true – almost all of the time. We see later in the chapter that this intuition is not always correct, and in this case, “almost always” needs to be entered into the statement. This scenario of two high-yield, deeply discounted bonds demonstrates a real bond math curiosity – the 30-year bond actually has lower sensitivity to a change in its yield to maturity than the 20-year bond. To understand this oddity, you need to explore the mathematics behind duration.