Duration and Convexity: Measuring Interest-Rate Sensitivity

What you will learn

• What Macaulay duration and modified duration measure, and how to calculate them.
• How to turn duration into a euro estimate of price risk, including DV01.
• Why convexity makes the simple duration estimate slightly wrong, and how to correct it.
• Which bond features increase or decrease interest-rate sensitivity.

Explanation

The ETF course (lesson "Bond ETFs and money-market funds") introduced interest-rate risk. This lesson gives you the tools to measure it.

Macaulay duration

Macaulay duration is the weighted average time, in years, until you receive a bond's cash flows. Each year is weighted by the share of the bond's price that comes from that year's cash flow (its present value divided by the price):

Macaulay duration = Σ [ t × PV(CF(t)) ] ÷ Price

A zero-coupon bond (a bond with no coupons, paying only the face value at maturity) has a Macaulay duration equal to its maturity. Any coupon bond has a shorter duration than its maturity, because some money arrives earlier.

Modified duration and DV01

Modified duration converts this time measure into price sensitivity:

Modified duration = Macaulay duration ÷ (1 + y) (for annual coupons)

It tells you the approximate percentage price change for a 1 percentage-point change in yield:

%ΔPrice ≈ − Modified duration × Δy

Traders also use DV01 ("dollar value of one basis point"; in euro markets the same idea is called PVBP, price value of a basis point). One basis point (bp) is 0.01 percentage points. DV01 = modified duration × price × 0.0001.

Convexity

The real price–yield relationship is a curve, not a straight line. Convexity measures that curvature. For normal (non-callable) bonds it is positive, which means prices rise a little more when yields fall than they fall when yields rise by the same amount. The improved estimate is:

%ΔPrice ≈ − D_mod × Δy + ½ × Convexity × (Δy)²

For small yield changes the convexity term is tiny; for large moves it matters.

Rules of thumb

• Longer maturity → higher duration.
• Lower coupon → higher duration (more of the value comes at the end).
• Lower yield level → higher duration.
• Portfolio duration ≈ the value-weighted average of the individual durations.
• Callable bonds (which the issuer may repay early) can have negative convexity: their upside is capped when yields fall.

Worked example

Hypothetical example: the 5-year, 3% annual-coupon bond from lesson 1, face €1,000, priced at €955.48 to yield 4%. Fees, spreads and taxes are excluded.

• Modified duration = 4.71 ÷ 1.04 = 4.53.
• DV01 = 4.53 × €955.48 × 0.0001 ≈ €0.43 per €1,000 face value per basis point.
• Convexity (calculated from the same cash flows) ≈ 25.6.

Now let yields jump by +1 percentage point (Δy = 0.01) and fall by −1 point.

The convexity term is ½ × 25.6 × 0.0001 = 0.13%. With it, the estimate matches the exact price almost perfectly. Note the asymmetry: the gain (+€44.52) is larger than the loss (−€42.07).

Common mistake

Practice activity

On paper: you hold a hypothetical portfolio of €6,000 in a bond fund with modified duration 2.0 and €4,000 in a long bond with modified duration 8.0. (a) What is the portfolio duration? (b) Estimate the euro change if all yields rise by 0.5 percentage points. (c) What is the modified duration of a 10-year zero-coupon bond yielding 3%?

Key takeaways

• Modified duration ≈ percentage price change for a 1-point yield change; multiply by your position to get euros.
• DV01 gives the euro change per basis point and is useful for comparing positions.
• Convexity improves the estimate and makes gains larger than losses for equal yield moves.
• Long maturities, low coupons and low yields mean higher interest-rate risk.

Next step

In the next lesson you will see how yields differ across maturities and what the shape of the yield curve can and cannot tell you.