FRM Exam Part I · Applying Duration, Convexity, and DV01

Convexity and Second-Order Price Approximation for FRM Part I

Updated 11 October 2026 · Fact-checked

Convexity measures how a bond's duration changes as yields change. Duration alone gives a straight-line price estimate. For larger yield moves, use %ΔP ≈ −D × Δy + ½ × C × (Δy)², with Δy in decimals. Positive convexity adds to gains and cushions losses. Negative convexity, as in callable bonds and MBS, does the opposite.

Understand Convexity and Second-Order Price Approximation

The price-yield curve of a plain bond is curved, not straight. Duration is the slope of that curve at the current yield. It gives a straight-line estimate of the price change. For a small yield move the line is close to the curve. For a large move the line drifts away from it.

Convexity measures the curvature of the price-yield curve. It is the second derivative of price with respect to yield, divided by price. Adding a convexity term to the duration estimate is a second-order (Taylor) approximation. It corrects most of the error from using the straight line alone.

For a normal bond, convexity is positive. The curve bends upward, so the true price is above the duration-only estimate whichever way yields move. When yields fall, the price rises by more than duration predicts. When yields rise, the price falls by less. This is why investors value convexity. Between two bonds with the same duration, the one with higher convexity does better in big moves.

Some securities have negative convexity. The main examples are callable bonds and mortgage-backed securities. When yields fall, the issuer is likely to call the bond or borrowers are likely to prepay. Price gains are capped, and the curve bends downward at low yields. Duration also shortens as yields fall and lengthens as yields rise. Convexity for these securities must be measured with effective convexity, which reprices the security after shifting the curve up and down, because the cash flows change with rates.

The convexity term is small for small moves. It is multiplied by (Δy)², so a 10 bp move gives almost nothing, while a 200 bp move gives a visible correction. That is why exams use convexity with large shocks.

Key formulas to remember

Second-order price approximation (percentage): ΔP ÷ P ≈ −D × Δy + ½ × C × (Δy)²; D is modified (or effective) duration, C is convexity, Δy is the yield change in decimals (100 bp = 0.01).
Second-order price approximation (price change): ΔP ≈ −D × P × Δy + ½ × C × P × (Δy)²; Use this when you need a currency amount. Multiply the percentage change by the starting price or position value.
Convexity from cash flows (annual compounding): C = (1 ÷ P) × Σ [ t × (t + 1) × CFt ÷ (1 + y)^(t + 2) ]; For m payments a year, use periods and the periodic yield y/m, then divide the result by m² to express convexity in years squared.
Effective duration: D_eff = (P₋ − P₊) ÷ (2 × P₀ × Δy); P₋ is the price after yield falls by Δy, P₊ after it rises by Δy.
Effective convexity: C_eff = (P₋ + P₊ − 2 × P₀) ÷ (P₀ × (Δy)²); Works for bonds with embedded options. A negative result signals negative convexity.
Sign rule: Convexity term = ½ × C × (Δy)² has the same sign as C; Because (Δy)² is always positive, the convexity term is positive for positive C whether yields rise or fall.

How to solve Convexity and Second-Order Price Approximation questions

Use this routine for any question that asks you to estimate a price or percentage change from duration and convexity, or to compute convexity from prices.

1Identify what is given: starting price P₀, duration (check that it is modified or effective, not Macaulay), convexity, and the yield change.
2Convert the yield change to decimals. 150 bp = 0.015. Keep its sign: a rise is positive, a fall is negative.
3Compute the duration term: −D × Δy. A yield rise gives a negative term, a fall gives a positive one.
4Compute the convexity term: ½ × C × (Δy)². Square Δy first, then multiply by C, then halve. This term has the sign of C.
5Add the two terms to get the approximate percentage price change. Multiply by P₀ if you need the dollar change, and add to P₀ if you need the new price.
6If prices are given for yield shifts instead, compute effective duration and effective convexity with the formulas above. Check whether the convexity is positive or negative.
7Sanity check: with positive convexity, the answer must be better than the duration-only estimate. A gain should be larger, a loss smaller. With negative convexity, the reverse.
8Match your answer to the options and watch units, such as percent versus decimals or price per 100 versus position value.

Quickest way: Shortcut: work in percent with Δy² as a fixed number

When to use it: Use it when the question gives duration, convexity and a round yield shock and asks for a percentage change or price estimate.

Learn the squares: 50 bp → 0.000025, 100 bp → 0.0001, 150 bp → 0.000225, 200 bp → 0.0004.
Duration term in percent: −D × (shock in bp) ÷ 100. For D = 6.2 and +150 bp, that is −6.2 × 1.5 = −9.3%.
Convexity term in percent: ½ × C × (squared shock) × 100. For C = 48 and 150 bp: 0.5 × 48 × 0.000225 × 100 = 0.54%.
Add them: −9.3% + 0.54% = −8.76%.
Eliminate options first: a positive-convexity answer must be closer to zero than the duration-only figure for a rise, and farther from zero in the gain direction for a fall.

Common mistakes in Convexity and Second-Order Price Approximation

Plugging basis points straight into the formula, for example Δy = 100 instead of 0.01.
Yield shocks are quoted in bp, and the convexity term squares the error.
Fix: Convert to decimals before you start. Write 100 bp = 0.01 next to the question.
Forgetting the ½ in front of the convexity term.
Candidates remember convexity as the second derivative but forget that the Taylor expansion divides it by 2.
Fix: Write the full formula first: −D × Δy + ½ × C × (Δy)². Never write the convexity term without the ½.
Subtracting the convexity term when yields rise.
Candidates link the sign to the direction of the yield move, as they do for duration.
Fix: The convexity term uses (Δy)², so it keeps the sign of C. For positive C it is added in both directions.
Assuming all callable bonds and MBS always have negative convexity.
The rule is taught as a headline fact.
Fix: Negative convexity appears mainly when yields are low enough that the call or prepayment option is near the money. At high yields the option is far out of the money and convexity can be positive. Say 'tends to' in your reasoning.
Using Macaulay duration in the price approximation.
Macaulay duration is the first duration candidates learn.
Fix: The approximation needs modified duration for a plain bond, or effective duration for a bond with options. Convert Macaulay using D_mod = D_Mac ÷ (1 + y/m).
Mixing convexity units, such as a convexity quoted as 0.70 when the formula needs 70.
Some sources divide convexity by 100 for presentation.
Fix: Check magnitudes. Convexity of an ordinary 10-year bond is typically in the tens or low hundreds. Use the figure that matches how the formula is written in the question.

Worked examples

Example 1

A bond is priced at 98.50 per 100 face value. Its modified duration is 6.2 and its convexity is 48. Yields rise by 150 bp. What is the best estimate of the percentage price change? (A) −9.30% (B) −8.76% (C) −8.22% (D) −9.84%

Show the solution

Convert the shock: Δy = +150 bp = 0.015.
Duration term: −D × Δy = −6.2 × 0.015 = −0.0930, or −9.30%.
Convexity term: ½ × C × (Δy)² = 0.5 × 48 × 0.015² = 0.5 × 48 × 0.000225 = 0.0054, or +0.54%.
Sum: −9.30% + 0.54% = −8.76%.
Check in price terms: 98.50 × −0.0876 = −8.63, so the new price is about 89.87.
Sanity check: with positive convexity the loss (8.76%) is smaller than the duration-only loss (9.30%). This is consistent.

Answer: (B) −8.76%, a price fall of about 8.63 to roughly 89.87.

Example 2

A callable bond trades at 100.00. If yields fall by 100 bp its price is 103.00. If yields rise by 100 bp its price is 96.50. Compute effective duration and effective convexity, and explain what the convexity tells you.

Show the solution

Δy = 0.01, P₀ = 100, P₋ = 103.00, P₊ = 96.50.
Effective duration = (P₋ − P₊) ÷ (2 × P₀ × Δy) = (103.00 − 96.50) ÷ (2 × 100 × 0.01) = 6.50 ÷ 2 = 3.25.
Effective convexity = (P₋ + P₊ − 2 × P₀) ÷ (P₀ × Δy²) = (103.00 + 96.50 − 200) ÷ (100 × 0.0001) = −0.50 ÷ 0.01 = −50.
Check with the approximation for −100 bp: −3.25 × (−0.01) + 0.5 × (−50) × 0.0001 = 0.0325 − 0.0025 = 0.03, so +3.0%, giving 103.00. This matches the input price.
Interpretation: the convexity is negative. The price gain from a yield fall (+3.0%) is smaller than the loss from an equal yield rise (−3.5%). The call option caps the upside.

Answer: Effective duration = 3.25; effective convexity = −50. Negative convexity means price appreciation is capped when yields fall, which is typical of a callable bond.

Exam tips

Always check whether the question gives duration in modified, Macaulay or effective form. Convert before applying the formula.
Under time pressure, compute the duration term and the convexity term separately and add them. This makes sign errors easy to spot.
If a question asks which bond benefits most from a large yield move at equal duration, the answer is the one with higher convexity. For negative convexity securities, expect statements about capped gains, such as in callable bonds and MBS.
For effective convexity, write down P₀, P₋ and P₊ before you calculate. Mixing up P₋ and P₊ reverses the duration sign.
Use your financial calculator or the memory function to hold Δy² when the same shock is used for several parts of a question.

Practice questions from Applying Duration, Convexity, and DV01

Convexity and Second-Order Price Approximation in other exams

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

Convexity and Second-Order Price Approximation: frequently asked questions

What is the convexity formula for a bond in FRM Part I?

Convexity is C = (1 ÷ P) × d²P/dy². For annual cash flows this is (1 ÷ P) × Σ t(t + 1) × CFt ÷ (1 + y)^(t + 2). For bonds with embedded options, use effective convexity: (P₋ + P₊ − 2P₀) ÷ (P₀ × Δy²).

How do I calculate a price change using duration and convexity?

Use %ΔP ≈ −D × Δy + ½ × C × (Δy)², with Δy in decimals. The first term is the duration estimate and the second is the convexity correction. Multiply the result by the starting price for the currency change.

What is negative convexity in callable bonds and MBS?

It means the price-yield curve bends downward, so price gains are limited when yields fall. In a callable bond the issuer may call the bond, and in an MBS borrowers may prepay. Duration then falls as yields fall and rises as yields rise.

Why does convexity matter for large yield changes?

Duration is a straight-line estimate that is accurate only for small moves. The bigger the yield change, the more the curvature matters, because the convexity term grows with the square of the move. Ignoring it understates gains and overstates losses for a positively convex bond.