Duration, Convexity and Interest Rate Risk

In the lesson on bonds you learned the most important rule of the bond market: when yields rise, prices fall – and duration tells you by how much. For everyday use this rule of thumb is enough. But if you want to understand why long bonds lost more than 40% in 2022, how insurers secure their promises or which bond ETF fits your own horizon, you need the maths behind it. It’s simpler than it sounds.

Macaulay duration: the average waiting time#

A bond is a sequence of payments: a coupon every year, and at the end the coupon plus face value. In 1938 the economist Frederick Macaulay asked: how long do you wait for your money on average – if each payment is weighted by its value today?

Take a bond with three years to maturity, a 5% coupon and a market yield of 5% too. It costs exactly $100.

The Macaulay duration is 2.86 years. It is slightly below the maturity because some of the money comes back earlier. Three simple rules follow:

• A zero-coupon bond has only one payment – its duration equals its maturity exactly.
• Higher coupons lower duration because more money comes back earlier.
• Higher yields lower duration too: late payments are discounted more heavily and count for less.

Modified duration: price sensitivity#

More interesting for investors is the question: by what percentage does the price move when yields change? For that you divide Macaulay duration by (1 + yield) and get the modified duration:

If the market yield rises one point to 6%, our bond’s price falls by about 2.72% by this calculation. The exact figure is 2.67% – the estimate is very good for small changes.

DV01#

Traders and banks don’t think in percent but in money. DV01 (“dollar value of one basis point”) tells you how much a position gains or loses when yields move by 0.01 percentage points:

With $1 million in our bond that is 2.72 × $1,000,000 × 0.0001 ≈ $272 per basis point. If yields rise 10 basis points in a day, the position loses about $2,720. DV01 lets you compare positions of different maturities directly and hedge them against each other.

Try it yourself: with a 0% coupon and 30 years to maturity you can see how extremely a long zero-coupon bond reacts to rates.

Convexity: why the curve bends#

Duration describes a straight line. The actual relationship between yield and price, however, is curved – as yields fall the price rises ever more steeply; as yields rise it falls ever more gently. This curvature is called convexity.

With convexity the estimate becomes more accurate:

For bondholders convexity is therefore a good thing – a kind of free bonus in big rate moves. Accordingly it has a price: bonds with high convexity yield a little less, all else being equal.

The limits of duration#

Duration assumes all yields shift at the same time by the same amount. In reality the yield curve twists and bends: short rates rise while long rates fall, or the other way round. Professionals therefore also measure key rate durations – the sensitivity to rate changes at individual points of the curve (2, 5, 10, 30 years).

Credit risk is not in duration either. A corporate bond can lose even as government yields fall if its credit spread widens.

Immunisation: neutralising interest rate risk#

Rising rates have two opposing effects for bondholders. The price falls immediately – but all future coupons can be reinvested at higher rates. When rates fall it’s the reverse. If the investment horizon equals the Macaulay duration, the two effects roughly cancel out. The result at the end of the horizon is then largely independent of how rates move.

Insurers and pension funds use this immunisation to secure future payment promises. For private investors it yields a practical rule:

Summary#

Macaulay duration is the present-value-weighted waiting time for a bond’s money, modified duration its price sensitivity and DV01 the same information in dollars. Convexity corrects the linear estimate and makes big rate cuts more valuable than equally big rate rises are harmful. And matching duration to your investment horizon largely turns interest rate risk into planning certainty.