FRM Exam Part I · Interest Rates

Duration and Convexity for FRM Part I

Updated 11 October 2026 · Fact-checked

Duration measures how sensitive a bond's price is to yield changes. Convexity corrects for the curvature that duration misses. Modified duration = Macaulay duration ÷ (1 + y/m). Estimate the price change as ΔP/P ≈ −D_mod × Δy + ½ × C × (Δy)². DV01 is the dollar price change for one basis point.

Understand Duration and Convexity

A bond's price falls when its yield rises. Duration tells you by roughly how much. It is the first-order (slope) measure of the price-yield curve. A modified duration of 6 means the price falls about 6% for a 1% (100 bp) rise in yield, for small moves.

Macaulay duration is the present-value-weighted average time to receive the bond's cash flows, measured in years. A zero-coupon bond's Macaulay duration equals its maturity. Coupons pull the average earlier, so a coupon bond's duration is shorter than its maturity. Modified duration converts this into a price sensitivity: Macaulay duration ÷ (1 + y/m), where y is the annual yield and m is the number of compounding periods per year. Dollar duration is modified duration × price. DV01 (dollar value of a basis point) is the price change for a 1 bp move: DV01 ≈ modified duration × price × 0.0001.

The price-yield curve is not a straight line. It bends upward (for a bond without embedded options). Duration is the tangent line, so it overstates the price fall when yields rise and understates the price gain when yields fall. Convexity measures that bend. Adding a convexity term makes the estimate closer to the true price change, especially for large yield moves.

For bonds with embedded options, such as callable bonds or mortgage securities, cash flows change when yields change. Use effective duration and effective convexity, which come from repricing the bond at yields shifted up and down. Convexity of an option-free bond is always positive. Callable bonds can show negative convexity.

Key formulas to remember

Macaulay duration: D_Mac = Σ [t × PV(CF_t)] ÷ P; t is in years. PV(CF_t) is the present value of each cash flow at the bond's yield. P is the bond price. For a zero-coupon bond, D_Mac = maturity.
Modified duration: D_mod = D_Mac ÷ (1 + y/m); y is the annual yield, m the compounding periods per year. For annual compounding, divide by (1 + y).
Duration price approximation: ΔP/P ≈ −D_mod × Δy; Δy is in decimals (50 bp = 0.005). Works for small yield changes.
Dollar duration and DV01: Dollar duration = D_mod × P; DV01 = D_mod × P × 0.0001; DV01 is the dollar change for a 1 bp yield move. Scale by the face amount held.
Convexity (annual cash flows): C = (1/P) × Σ [t × (t + 1) × CF_t ÷ (1 + y)^(t + 2)]; With m payments per year and t counted in periods, divide the result by m² to express it in years squared.
Duration plus convexity approximation: ΔP/P ≈ −D_mod × Δy + ½ × C × (Δy)²; The convexity term is always positive when C is positive, so it adds to the price change.
Effective duration: D_eff = (P₋ − P₊) ÷ (2 × P₀ × Δy); P₋ and P₊ are prices after the yield falls and rises by Δy. Used for bonds with embedded options.
Effective convexity: C_eff = (P₋ + P₊ − 2 × P₀) ÷ (P₀ × (Δy)²); Same shifted prices as effective duration.
Portfolio duration: D_p = Σ w_i × D_i; w_i are market-value weights. DV01s of positions simply add.

How to solve Duration and Convexity questions

Use this order for any duration or convexity question. It keeps units and signs under control.

1Read what is given: price, yield, compounding, cash flows, and whether duration is Macaulay, modified or effective.
2Convert the yield change to decimals. 25 bp = 0.0025. Note the sign: a yield rise gives a price fall.
3If you need modified duration and only have Macaulay, divide by (1 + y/m) using the right m.
4Compute the first-order effect: −D_mod × Δy for a percentage change, or −D_mod × P × Δy for a dollar change. For DV01, use Δy = 0.0001.
5If the move is large or the question mentions convexity, add ½ × C × (Δy)². Check that C and Δy use matching units.
6Combine the two terms, then convert the percentage into a price or a position value.
7Sanity check: the convexity term should be small and positive. The estimate should sit slightly above the duration-only estimate.

Quickest way: Dollar duration shortcut

When to use it: Use this when the question gives duration and market value and asks for a price change, DV01 or hedge size. You do not need cash flows.

Compute DV01 = D_mod × market value × 0.0001.
Multiply DV01 by the move in basis points for the first-order dollar change.
Add ½ × C × (Δy)² × market value only if convexity is given and the move is large (roughly 50 bp or more).
For a hedge, divide the position DV01 by the hedging instrument's DV01 to get the number of contracts.
Pick the answer with the correct sign. Yields up means value down.

Common mistakes in Duration and Convexity

Using Macaulay duration directly in the price-change formula.
Both are called duration and are often close in value.
Fix: Divide Macaulay duration by (1 + y/m) first. The price-change formulas use modified duration.
Entering Δy in percent or basis points instead of decimals.
Yield moves are quoted as '50 bp' or '0.5%', and students plug in 50 or 0.5.
Fix: Convert to decimals first: 50 bp = 0.005. Check that a 100 bp move gives a price change near the duration number in percent.
Forgetting the sign, or subtracting the convexity term.
The first term is negative for a yield rise, and students apply the minus sign to both terms.
Fix: The convexity term ½ × C × (Δy)² is positive regardless of the direction of the yield move, because Δy is squared.
Using the wrong compounding in modified duration or convexity.
A semiannual bond has duration in half-year periods that must be converted to years.
Fix: State the periods per year. Divide Macaulay duration by (1 + y/m) with the correct m. If you compute convexity in periods, divide by m².
Applying duration and convexity to a bond with embedded options.
The formulas look the same for any bond.
Fix: When cash flows depend on yield, use effective duration and convexity from repriced bonds.
Treating duration as exact for large yield moves.
The formula looks precise, so students trust it.
Fix: Duration is a linear estimate. For large moves the error grows, and convexity reduces it but does not remove it.

Worked examples

Example 1

A 3-year bond pays an annual coupon of 6% on a face value of 100 and has a yield of 6% (annual compounding), so its price is 100. Find the Macaulay duration, the modified duration, the convexity, and the estimated percentage price change if the yield rises by 50 bp, using duration and convexity.

Show the solution

Cash flows: 6 at t = 1, 6 at t = 2, 106 at t = 3.
Present values at 6%: 6 ÷ 1.06 = 5.6604; 6 ÷ 1.1236 = 5.3400; 106 ÷ 1.191016 = 88.9996. The total is 100.00, which confirms the price.
Macaulay duration = (1 × 5.6604 + 2 × 5.3400 + 3 × 88.9996) ÷ 100 = (5.6604 + 10.6800 + 266.9988) ÷ 100 = 2.8334 years.
Modified duration = 2.8334 ÷ 1.06 = 2.6730.
Convexity terms t(t + 1) × CF ÷ (1.06)^(t + 2): t = 1: 12 ÷ 1.191016 = 10.075; t = 2: 36 ÷ 1.262477 = 28.514; t = 3: 1,272 ÷ 1.338226 = 950.52. The sum is 989.11, and C = 989.11 ÷ 100 = 9.891.
Duration term: −2.6730 × 0.005 = −0.013365, or −1.3365%.
Convexity term: ½ × 9.891 × (0.005)² = 0.5 × 9.891 × 0.000025 = 0.000124, or +0.0124%.
Total: −1.3365% + 0.0124% = −1.3241%.

Answer: Macaulay duration ≈ 2.833 years, modified duration ≈ 2.673, convexity ≈ 9.89. The estimated price change is about −1.32%, so the price falls to about 98.68.

Example 2

A portfolio has a market value of $50 million and a modified duration of 6.2. (a) Find the portfolio DV01. (b) Estimate the dollar loss if yields rise 25 bp, ignoring convexity. (c) A futures contract has a DV01 of $850. How many contracts are needed to hedge the portfolio's DV01, rounded to the nearest whole contract?

Show the solution

(a) DV01 = 6.2 × 50,000,000 × 0.0001 = $31,000 per basis point.
(b) Loss = DV01 × 25 = 31,000 × 25 = $775,000. Check: −6.2 × 0.0025 = −1.55%, and 1.55% × 50,000,000 = $775,000.
(c) The portfolio loses value when yields rise, so you need a hedge that gains. Take a short futures position. Contracts = 31,000 ÷ 850 = 36.47, which rounds to 36.

Answer: DV01 = $31,000. Estimated loss for +25 bp ≈ $775,000. Hedge with about 36 short futures contracts.

Exam tips

Know the difference between Macaulay, modified and effective duration. Questions often test which one belongs in a given formula.
Always convert basis points to decimals before calculating. Many wrong options are built from the 100× slip.
Expect a conceptual question on convexity: positive convexity helps the holder, and the duration-only estimate overstates losses and understates gains.
For DV01 and hedging questions, compute the DV01 of both positions in dollars first, then take the ratio.
On a financial calculator, get the price with N, I/Y, PMT, FV and CPT PV. Reprice at yields ±Δy to check effective duration and convexity.

Practice questions from Interest Rates

Duration and Convexity in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Duration and Convexity: frequently asked questions

What is the difference between modified duration and Macaulay duration?

Macaulay duration is a weighted average time to cash flows, in years. Modified duration is Macaulay duration divided by (1 + y/m), and it measures the percentage price change for a small yield change. Use modified duration for price-sensitivity questions.

How do you calculate DV01 for FRM Part I?

DV01 = modified duration × market value × 0.0001. It gives the dollar price change for a one basis point change in yield. You can also compute it by repricing the bond at yields 1 bp apart.

Why is convexity added to the duration estimate?

The price-yield curve is curved, but duration is a straight-line approximation. The convexity term ½ × C × (Δy)² corrects for the curvature, which matters more for larger yield changes.

Can convexity be negative?

Yes. Option-free bonds have positive convexity. Bonds with embedded options, such as callable bonds and many mortgage securities, can show negative convexity over some yield ranges, so use effective measures for them.