Duration: how strongly bonds react to interest rates

In depth9 min read · 8 sources

Duration tells you how strongly a bond's price reacts to changes in interest rates: with a duration of 8, it loses about 8% if rates rise by one percentage point. It is the most important metric for bond ETFs and for any portfolio in which bonds are meant to reduce risk. This article explains the variants, from Macaulay duration and convexity to key rate and spread duration.

Macaulay duration and modified duration

In his 1938 study (Macaulay 1938), Frederick Macaulay defined duration as the average time to payment: the average of a bond's payment dates, weighted by the present value of each payment. Dividing it by (1 + yield) gives the modified duration, i.e. the percentage price change per percentage point change in interest rates.

The longer the maturity and the lower the coupon, the higher the duration. For a zero-coupon bond, the Macaulay duration equals the maturity, because there is only one payment. A bond ETF has the weighted duration of its holdings: based on the holdings lists of October 2026, a euro government bond ETF covering all maturities comes out at around 6.7, and a euro corporate bond ETF at around 4.4.

Convexity: why the approximation is not enough

A bond's price does not depend linearly on the interest rate; the relationship is curved. Convexity measures this curvature and improves the approximation:

Positive convexity is pleasant for investors: if rates rise, the bond loses slightly less than the duration suggests; if they fall, it gains slightly more. Callable bonds or mortgage-backed securities, by contrast, often have negative convexity: when rates fall, they are repaid early, and the price gain remains limited. For them, an effective duration is quoted, derived from a valuation model with and without a shift in rates.

Key rate duration: interest rate risk by maturity

Interest rates for different maturities do not always move in step. The key rate duration of Ho (1992) distributes the duration across maturity points (1, 2, 3, 5, 7, 10, 15, 20, 30 years): how much does the portfolio lose if only the 10-year rate rises? Together, the key rate durations add up to the duration.

Why are a few maturity points enough? Litterman and Scheinkman (1991) showed that three common factors (level, slope and curvature of the yield curve) explain almost all movements in bond yields. Similarly, central banks model the entire curve with a few parameters, following Nelson and Siegel (1987) and the extension by Svensson (1994). Key rate durations reveal how a portfolio reacts to a steeper or flatter curve, not just to a parallel shift.

Spread duration: the risk of rising risk premiums

Corporate, covered and emerging market bonds pay a premium (spread) over safe government bonds. If it widens in a crisis, prices fall even if safe interest rates stay the same. Spread duration measures exactly that; for fixed-coupon bonds, it is almost equal to the interest rate duration. Floating-rate bonds have hardly any interest rate duration, but they do have spread duration. A corporate bond ETF with a spread duration of 4.4 therefore loses around 4.4% if spreads rise by one percentage point.

Using duration in a portfolio

The duration of a portfolio is the weighted average: 60% in a bond ETF with a duration of 6.7 and 40% in the money market at 0.2 give 0.6 × 6.7 + 0.4 × 0.2 = 4.1. Expressed in euros, the metric is called DV01 (change in value per basis point): for €10,000 in the 10-year bond, this is 10,000 × 8.53 × 0.0001 = €8.53 per basis point.

• Protection in recessions: long-dated government bonds gain when interest rates fall, and this often happens when equities are losing.
• Inflation risk: if interest rates rise because of inflation, equities and bonds lose at the same time (2022).
• Hedging goals: a payment due in seven years can be hedged against interest rate changes with bonds of similar duration. This is the principle behind asset-liability management.
• Watching the yield curve: where the key rate durations lie determines whether a portfolio benefits from a flattening or a steepening yield curve.

Ladder, barbell or bullet

The way a bond holding is spread across maturities has its own names. A ladder invests equal amounts in each maturity, for example from one to ten years; whatever matures is reinvested at the long end. A barbell combines very short and very long bonds, while a bullet concentrates around one maturity. At the same duration, the barbell is more convex than the bullet: with large changes in rates, it gains slightly more and loses slightly less (Tuckman and Serrat 2012). In return, its result depends on the shape of the yield curve: if long rates fall and short rates rise (the curve flattens), the barbell comes out ahead; if the curve steepens, the bullet tends to.

Combining yield and rate changes

For a one-year holding period, the return of a bond ETF can be roughly split into two parts: the running yield you get if rates stay unchanged, and the price change caused by a change in rates. The rule of thumb leaves out small effects such as convexity and rolling down the curve.

The calculation also shows why long bonds were so vulnerable in low-interest periods: with a yield close to zero and a duration of 15, a small rise in rates is enough for a double-digit loss. Conversely, higher yields offer a buffer that offsets price losses more quickly.

Advanced: immunisation

Redington (1952) formulated the principle of immunisation for life insurers: if assets and liabilities have the same present value and the same duration, a small parallel shift in rates affects both equally. If, in addition, the convexity of the assets is greater than that of the liabilities, every shift even produces a small surplus. Using US bond data, Fisher and Weil (1971) showed that a strategy with matching duration reaches the target much more reliably than one with matching maturity.

• Duration drifts: over time and as rates change, the duration of assets and that of liabilities change at different speeds. Immunisation requires regular readjustment.
• Parallel shifts only: a single duration does not protect against twists in the curve; for that, the key rate durations must match.
• Real or nominal rates: if you want to protect purchasing power, you need inflation-linked bonds; duration then measures the risk of the real interest rate.

How Portfolio Lab calculates

The “Risk factors” tab in the Look-through takes the duration of each bond from the provider's holdings list. If it is missing, Portfolio Lab calculates the modified duration from maturity and coupon as for a bond with a fixed annual coupon; the yield comes from the file, otherwise the coupon is used, and without a coupon 3%. Key rate durations are obtained by distributing each payment, according to its contribution to duration (time × present value), linearly between the two neighbouring maturity points from 1 to 30 years. The total is the duration. Spread duration counts the duration of all bonds except government bonds in the home currency; cash and money market positions carry no spread. Funds without holdings data are estimated from their maturity segment and flagged accordingly. More in the risk factors methodology.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.