Duration and Convexity

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9-205-025

REV: SEPTEMBER 27, 2004

GEORGE CHACKO

PETER HECHT

VINCENT DESSAIN

ANDERS SJÖMAN

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Note on Duration and Convexity

When interest rates go up, bond prices fall.

All actors in the financial markets—money managers, traders, arbitrageurs—know this. But how

do they compare how much prices will change between various financial instruments, before interest

rates actually change? To inform their strategies, traders want to know, for instance, which bond will

lose more in price given that interest rates rise 100 basis points: a 6% coupon bond with a 15-year

maturity or one with a 30-year maturity—or maybe a 9% coupon bond with a 30-year maturity. To

help estimate price sensitivity between bonds of different maturities and coupon rates, two measures

are normally used: duration and convexity.

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As measures, however, duration and convexity are not just limited to bonds. They apply to all

financial instruments, fixed income and equity alike. They also measure price reaction to risk factors

other than just interest rates. Therefore, put more generally, duration and convexity take into account

any change for any risk factor affecting the price of any financial instrument. Although many

examples in this note will center on their use for bonds regarding yield and interest rate changes, this

larger application of duration and convexity should not be underrated.

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The main difference between the two concepts is that duration focuses on small changes in risk

factors, and convexity then builds on duration to adjust for larger changes. For bonds, for instance,

the basic price-yield relationship of an option-free bond—a bond without any embedded options,

such as being callable or putable—dictates that the bond price will change in the opposite direction of

the change in the required yield.1 (A more formal restatement of this note’s first sentence.) When

graphed, this correlation appears as a convex curve (see...