FI 12 Yield-Based Bond Convexity and Portfolio Properties

Modified duration measures how a bond price responds to a yield change as though that relationship were a straight line. In reality the link between a bond price and its yield-to-maturity traces a curved path, so a purely linear measure loses accuracy once yields move by a large amount. Convexity is the second-order measure that captures this curvature for an option-free fixed-rate bond, and it is layered on top of the duration estimate to sharpen the prediction. The larger the yield move, and the longer the maturity, the more the curvature matters.

For an option-free fixed-rate bond, convexity is always positive. That sign carries a welcome consequence for the holder. When yields fall, the actual price rises by more than duration alone predicts; when yields rise, the actual price falls by less than duration alone predicts. Because the correction runs in the investor’s favour in both directions, greater convexity is a desirable property. In a competitive market investors therefore pay for it: a bond whose extra convexity is recognised will carry a higher price and, equivalently, a lower yield-to-maturity.

The drivers of convexity line up with those of duration. Convexity is greater for a fixed-rate bond whose time-to-maturity is longer, whose coupon rate is lower, and whose yield-to-maturity is lower. One additional driver is the dispersion of the cash flows, meaning how widely the payments are spread across time. Given equal duration, whichever bond has payments spread more widely across time carries the greater convexity.

Two further ideas complete the toolkit. Money convexity restates convexity in currency (or percent of par) terms for an actual position: it is the annual convexity multiplied by the full price of the position. And for a group of bonds, portfolio duration and convexity can be found either by discounting the aggregate cash flows directly, which is theoretically correct but awkward in practice, or by taking market-value weighted averages of the individual bond measures, which is the method managers actually use. The rest of this lesson develops each of these in turn.

Modified duration captures the first-order (linear) effect of a yield change on price. Convexity captures the second-order (non-linear) effect for an option-free fixed-rate bond. Put the two together and the estimated price change tracks the true price change closely, even for a sizeable yield move. The combined estimate for the percentage full-price change is Equation 1.

Computing convexity from the cash flows

Annual convexity can be built the same way as duration, with one extra column. For each period, multiply the time to receipt by that time plus one, by the present-value weight of the cash flow, and by a discounting adjustment, as shown below. Multiplying the time by the time plus one introduces the squared-time term that gives convexity its non-linear character.

Consider BRWA Corporation’s 3.2% semiannual coupon bond, which has five years to maturity and trades at par, settling 15 October 2025 and due 15 October 2030 (principal USD300,000,000, yield-to-maturity 3.20%). The table works the ten semiannual periods.

The last column sums to 96.9558. Because there are two periods per year, divide by two squared (that is, by four) to annualize: 96.9558 / 4 = 24.2389. So the annualized Macaulay duration is 4.6601 and annualized convexity is 24.2389.

Approximating convexity from three prices

When cash flows are uncertain, for example with embedded options or default risk, convexity can be approximated from three full prices: the base price and the prices after a small equal rise and fall in yield. Equation 2 uses the same inputs as the approximation for modified duration.

Why greater convexity helps

Take two bonds that match on price, on maturity, on yield-to-maturity, and on modified duration; at the current yield they therefore rest on one common tangent line. The one with more convexity gains more when yields drop and loses less when yields climb. It therefore outperforms the less convex bond whether yields fall or rise, which makes it the less risky holding. This advantage assumes the extra convexity is not already reflected in a higher price; if it were priced in, the more convex bond would simply trade richer at a lower yield.

With convexity in hand, the estimated percentage price change for any yield move follows directly from Equation 1: subtract the duration effect, then add the convexity adjustment. To see how much the adjustment buys, return to BRWA’s five-year 3.2% bond and move its yield by 100 bps each way from 3.20%. The exact repriced values are PV_+ = 95.53212 at 4.20% and PV_- = 104.71035 at 2.20%, so the true percentage changes are a fall of 4.46788% and a rise of 4.71035%.

Modified duration for this bond is 4.58676, so duration alone predicts changes of exactly −4.58676% and +4.58676%, symmetric by construction. Adding the convexity term (annual convexity 24.23895) breaks that symmetry in the right way.

Duration alone misses the true change by roughly 12 bps in each direction, while duration plus convexity lands within a quarter of a basis point. The gap would widen further for larger yield moves or for longer-maturity, lower-coupon bonds.

Money duration and money convexity

To express the same idea in currency, money duration gives the first-order price change in currency units, and money convexity gives the second-order piece. Money convexity is the annual convexity multiplied by the full price of the position.

Duration and convexity describe the interest rate risk of a portfolio just as they do for a single bond. There are two ways to obtain them. The first discounts the portfolio’s aggregate cash flows and takes the weighted average of their times to receipt; this is theoretically correct but cumbersome. The second takes market-value weighted averages of the individual bonds’ durations and convexities. Portfolio managers rely on the second method, so it is the focus here, along with its main limitation.

Consider a two-bond portfolio holding USD50 million par each of BRWA’s five-year bond and the 30-year government of Romania bond, both quoted in US dollars.

Weighting each bond’s statistic by its share of portfolio market value gives the portfolio measures.

These feed straight into Equation 1. For a 100 bp rise in the yields of both bonds, the portfolio is estimated to lose 9.21396% of its value.

The weighted-average approach is simple and quick, and it grows more accurate as the bonds’ yields-to-maturity converge and the yield curve flattens. Its limitation is the assumption baked into it: the measures implicitly assume that yields at every maturity move together by an equal amount, which is a parallel shift of the curve. Real yield curves rarely move that cleanly; steepening, flattening, and twisting are far more common.