Skip to main content
Other Assets

Chapter 4: Duration & Convexity (Gentle)

Pomegra Learn

Chapter 4: Duration & Convexity (Gentle)

When you own a bond, two things change its value: the passage of time (which inexorably brings you closer to maturity and coupon payments) and movements in interest rates (which change how much other investors will pay for those future cash flows).

Duration and convexity are the tools that quantify these effects. Duration tells you how much a bond's price changes when rates move. Convexity tells you why that relationship isn't perfectly straight-line—why longer-duration bonds benefit more from falling rates than they lose from rising rates, a phenomenon that has shaped bond markets for decades.

This chapter builds from first principles. We start by asking: what does it mean to "wait" for a bond's cash flows, and how long is the average wait? That's duration. We explore three equivalent views of duration—as a time-weighted average, as interest rate sensitivity, and as a formula—showing that all three perspectives describe the same economic reality.

Then we examine the relationships that drive duration: how coupon size, maturity length, and yield levels all pull duration in different directions. We show why a high-coupon bond feels less risky than a low-coupon bond of the same maturity (shorter duration), and why a bond's sensitivity to rates isn't simply "how many years until maturity?"

Finally, we introduce convexity—the subtle but powerful curvature in the price-yield relationship that emerges from the mathematics of discounting. Convexity is why an investor prefers bonds to many other assets in uncertain rate environments: it's free optionality.

This chapter assumes you understand bond basics (coupon, maturity, yield to maturity, the inverse price-yield relationship) and are ready to move beyond intuition into the quantitative foundation of bond analysis. By the end, you'll have the conceptual and mathematical tools that professional bond managers use daily to measure risk, construct portfolios, and exploit opportunities.

What's in this chapter​

How to read it​

Start with "What Is Duration?" if you're new to the concept. It establishes the intuition: duration is when you get your money back, weighted by cash flow size and timing.

Move through "Macaulay Duration" and "Modified Duration" if you want the mathematical foundations. Macaulay duration is the original 1938 measure; modified duration adjusts it for the yield effect, producing the direct price-sensitivity formula traders use.

The middle section ("Duration as Time-Weighted Average," "Duration as Price Sensitivity," "Effective Duration") offers different perspectives on the same calculation. All describe duration; each perspective illuminates a different use case. Professional investors naturally shift between them.

The final section—"Zero-Coupon," "Coupon Bonds," "Portfolio," and the three "Relationships" articles—explores how duration varies with bond characteristics. This is where you'll develop intuition for why a high-coupon bond reacts differently to rates than a low-coupon bond, why 30-year bonds don't have 30-year duration, and how portfolio managers construct specific interest rate exposures.

If you're in a hurry, "What Is Duration?" plus one of the "Relationships" articles gives you the core concepts. If you're building a professional toolkit, read the entire chapter sequentially; each builds on earlier material.

Duration is universal across all bond types—Treasuries, corporates, municipals, high-yield, international bonds. Once you master it here, you can apply the same concepts to any fixed-income instrument.