Introduction

Duration and convexity are core tools for measuring interest-rate risk. They explain not only whether a bond is rate-sensitive, but approximately how much its price may move when yields change.

What Is Duration?

Duration is a family of measures used to describe the timing of bond cash flows and, more importantly for risk management, sensitivity to changes in yield. It converts a complex stream of coupons and principal into a compact risk measure.

Duration should not be interpreted simply as maturity. Two bonds with the same maturity can have different durations because their coupons and yields differ.

Macaulay Duration

Macaulay duration is the present-value-weighted average time at which a bond’s cash flows are received. Each cash flow is weighted by its contribution to the bond’s current price.

A zero-coupon bond’s Macaulay duration equals its maturity because its only cash flow occurs at maturity. A coupon-paying bond normally has a shorter Macaulay duration because some value is received earlier.

Modified Duration

Modified duration measures the sensitivity of a bond’s price to changes in yield. More specifically, it estimates the percentage change in the bond’s price resulting from a small change in yield.

The relationship is expressed as:

\[
\frac{\Delta P}{P} \approx -D_{\mathrm{mod}} \times \Delta y
\]

where:

* \(P\) is the current bond price;
* \(\Delta P\) is the change in the bond price;
* \(D_{\mathrm{mod}}\) is the modified duration;
* \(\Delta y\) is the change in yield expressed in decimal form.

The negative sign reflects the inverse relationship between bond prices and yields: when yields rise, bond prices generally fall, and when yields fall, bond prices generally rise.

For example, suppose a bond has a modified duration of 6 and its yield increases by 0.50 percentage points. Since 0.50% equals 0.005 in decimal form, the estimated price change is:

\[
\frac{\Delta P}{P} \approx -6 \times 0.005 = -0.03 = -3.0\%
\]

Therefore:

\[
\frac{\Delta P}{P} \approx -3.0\%
\]

The bond’s price would therefore be expected to decline by approximately 3.0%.

Modified duration provides a first-order, or linear, approximation of price sensitivity. It is generally useful for relatively small changes in yield. For larger yield movements, the estimate becomes less accurate because the relationship between bond prices and yields is curved rather than linear. This curvature is captured by convexity, which can be combined with modified duration to improve the estimate of a bond’s price change.

What Is Convexity?

Duration assumes a locally linear relationship between a bond’s price and yield. In reality, however, the price-yield relationship is curved. Convexity measures this curvature and can be used together with modified duration to provide a more accurate estimate of price changes, particularly when yield movements are relatively large.

The duration-and-convexity approximation is:

\[
\frac{\Delta P}{P} \approx -D_{\text{mod}} \Delta y + \frac{1}{2} \times \text{Convexity} \times (\Delta y)^2
\]

where:

* \(P\) is the current bond price;
* \(\Delta P\) is the change in the bond price;
* \(D_{\text{mod}}\) is the modified duration;
* \(\Delta y\) is the change in yield expressed in decimal form;
* \(\text{Convexity}\) measures the curvature of the price-yield relationship.

The first term estimates the price change using modified duration, while the second term adjusts that estimate for convexity. For a traditional option-free bond with positive convexity, this adjustment is positive whether yields rise or fall because the change in yield is squared.

The exact definition and scaling of convexity may vary depending on the market convention and calculation methodology used.

Why Convexity Matters

For plain bonds with positive convexity, the price gain from a fall in yields tends to be slightly larger than the price loss predicted by a linear duration estimate, while the loss from a rise tends to be slightly smaller.

Embedded options can materially change this behavior. Callable bonds, for example, can exhibit negative convexity in some yield ranges because the issuer becomes more likely to call the bond when rates fall.

Portfolio Use

Portfolio managers use duration to set interest-rate exposure, compare securities and hedge risk. A portfolio can be positioned with shorter duration when the objective is to reduce sensitivity to rising yields, or longer duration when the investor deliberately wants more rate exposure.

Convexity helps distinguish portfolios with similar duration but different behavior under larger yield moves.

Key Takeaways

Macaulay duration focuses on the timing of cash flows. Modified duration estimates price sensitivity. Convexity captures curvature. Together they form a core framework for measuring and managing bond interest-rate risk.

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