FRM Exam Part II · Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

Duration and Convexity of Assets and Liabilities

Updated 11 October 2026 · Fact-checked

Duration measures how sensitive a bond's price is to a small change in yield. Convexity corrects that estimate for the curvature of the price-yield curve. To solve: ΔP/P ≈ −D_mod × Δy + ½ × C × (Δy)². Use effective duration when cash flows change with rates, as with callable bonds.

Understand Duration and Convexity of Assets and Liabilities

A bond price is the present value of its cash flows. When yields rise, the present value falls. The price-yield relationship is not a straight line. It is a curve that bends upward. Duration is the slope of that curve. Convexity is how fast the slope changes.

Macaulay duration is the weighted average time, in years, to receive the cash flows. Each weight is the present value of that cash flow divided by the bond price. A zero-coupon bond has a Macaulay duration equal to its maturity. A coupon bond has a shorter one.

Modified duration converts Macaulay duration into a price sensitivity: D_mod = D_Mac ÷ (1 + y/m), where y is the yield and m is the compounding frequency. It tells you the approximate percentage price change for a 1 unit change in yield. A modified duration of 6 means roughly a 6% price fall for a 1 percentage point (100 bp) rise in yield.

Duration is a linear estimate, so it is accurate only for small yield moves. For larger moves it understates the price rise when yields fall and overstates the price fall when yields rise, for a standard bond with positive convexity. Adding the convexity term fixes most of this error.

For bonds with embedded options, such as callable bonds or mortgage-backed securities, cash flows depend on rates. Modified duration assumes fixed cash flows, so it misleads. Effective duration reprices the bond under a downward and an upward yield shift using a model, and then measures the slope. A callable bond has lower effective duration than its modified duration suggests, and it can show negative convexity when yields are low.

For a portfolio, duration is the market-value-weighted average of the individual durations. For a bank, the same logic applies to assets and liabilities. The duration of equity depends on both sides and on leverage, which links to duration gap analysis.

Key formulas to remember

Macaulay duration: D_Mac = Σ [t × PV(CF_t)] ÷ P; t is time in years. PV(CF_t) is the present value of the cash flow at time t. P is the bond price.
Modified duration: D_mod = D_Mac ÷ (1 + y/m); y is the annual yield and m the number of coupons per year. For annual compounding, divide by (1 + y).
Duration approximation of price change: ΔP/P ≈ −D_mod × Δy; Δy in decimals. A 25 bp rise is 0.0025. Valid for small, parallel yield changes.
Duration plus convexity approximation: ΔP/P ≈ −D_mod × Δy + ½ × C × (Δy)²; C is convexity. Use the same compounding basis for D_mod and C.
Effective duration: D_eff = (P₋ − P₊) ÷ (2 × P₀ × Δy); P₋ is the price after yields fall by Δy, P₊ after yields rise by Δy, P₀ the starting price.
Effective convexity: C_eff = (P₋ + P₊ − 2 × P₀) ÷ (P₀ × (Δy)²); Same shifts as effective duration. Can be negative for callable bonds and mortgage securities.
Portfolio duration: D_p = Σ w_i × D_i; w_i is the market-value weight. Convexity of a portfolio is weighted the same way.
Dollar duration (DV01 link): DV01 ≈ D_mod × P × 0.0001; The price change in currency for a 1 bp move in yield.

How to solve Duration and Convexity of Assets and Liabilities questions

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

1Identify the measure asked for: Macaulay, modified, effective, or a price change. Note the compounding frequency.
2Check whether the bond has embedded options. If cash flows change with rates, use effective duration and convexity.
3Convert yield change to decimals. 50 bp becomes 0.005.
4If given Macaulay duration, compute modified duration by dividing by (1 + y/m).
5Compute the duration effect: −D_mod × Δy.
6Add the convexity effect ½ × C × (Δy)² if convexity is given or the move is large.
7Multiply the total percentage change by the starting price to get the currency change. Add it to the price for the new price.
8Sanity check the sign: rising yields cut prices. Convexity adds a positive amount for a standard bond.

Quickest way: Duration-first shortcut

When to use it: Use when the yield move is 50 bp or less, or when the options are far apart in value.

Compute −D_mod × Δy as a percentage. Example: 7 × 0.5% = 3.5%.
Add the convexity bump only if the options are close. Convexity effect in percent = 0.5 × C × (Δy in decimals)² × 100.
For portfolios, take the weighted average duration first, then apply it once.
Eliminate options with the wrong sign before calculating exactly.

Common mistakes in Duration and Convexity of Assets and Liabilities

Using Macaulay duration directly in the price change formula.
Both are measured in years, so they look interchangeable.
Fix: Divide Macaulay duration by (1 + y/m) first. Only modified duration gives percentage price sensitivity.
Forgetting the sign, so a yield rise gives a price gain.
The negative sign in −D × Δy is dropped when rushing.
Fix: State the direction before calculating. Yield up means price down. Convexity is the only positive term.
Plugging basis points straight into the formula.
Δy of 50 is used instead of 0.005.
Fix: Convert to decimals first. The convexity term squares Δy, so the error becomes very large.
Using modified duration for a callable bond.
The formula is familiar, and embedded options are overlooked.
Fix: For callable bonds and mortgage securities use effective duration from repricing in up and down scenarios.
Applying the convexity term as C × (Δy)² without the ½.
The factor ½ comes from the Taylor expansion and is easy to forget.
Fix: Write the full formula before substituting numbers: ΔP/P ≈ −D × Δy + ½ × C × (Δy)².
Averaging portfolio duration with par values or equal weights.
Positions look similar in size, or weights are taken from face value.
Fix: Weight by market value. Use dollar duration if positions include leverage or short positions.

Worked examples

Example 1

A bond priced at USD 98.00 has a modified duration of 7.2 and convexity of 70. Yields rise by 100 bp. Estimate the new price using duration and convexity.

Show the solution

Δy = 0.01.
Duration effect = −7.2 × 0.01 = −0.072, or −7.2%.
Convexity effect = ½ × 70 × (0.01)² = 0.5 × 70 × 0.0001 = 0.0035, or +0.35%.
Total change = −7.2% + 0.35% = −6.85%.
Price change = 98.00 × −0.0685 = −6.713.
New price = 98.00 − 6.713 = 91.287.

Answer: About USD 91.29 (a fall of 6.85%). Duration alone would give USD 90.94.

Example 2

A callable bond trades at 102.00. If yields fall 50 bp its model price is 103.20. If yields rise 50 bp its model price is 100.60. Calculate effective duration and effective convexity.

Show the solution

Δy = 0.005, P₀ = 102.00, P₋ = 103.20, P₊ = 100.60.
Effective duration = (103.20 − 100.60) ÷ (2 × 102.00 × 0.005) = 2.60 ÷ 1.02 = 2.549.
Effective convexity = (103.20 + 100.60 − 2 × 102.00) ÷ (102.00 × 0.005²).
Numerator = 203.80 − 204.00 = −0.20.
Denominator = 102.00 × 0.000025 = 0.00255.
Convexity = −0.20 ÷ 0.00255 = −78.4.

Answer: Effective duration is about 2.55 and effective convexity is about −78.4. The negative convexity is consistent with a callable bond, because the call option caps the price gain when yields fall.

Exam tips

Read the compounding basis. A semiannual bond uses (1 + y/2) in the modified duration formula.
When an option set includes a Macaulay-based answer and a modified-based answer, check which one the question needs.
Questions on callable bonds and MBS usually test effective duration and negative convexity, not the arithmetic.
Know the direction of errors: duration alone understates price gains and overstates price losses for positively convex bonds.
For portfolio questions, weight by market value and remember convexity is also a weighted average.

Practice questions from Risk Management for Changing Interest Rates: Asset-Liability Management and Duration Techniques

Duration and Convexity of Assets and Liabilities 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 of Assets and Liabilities: frequently asked questions

What is the difference between Macaulay duration and modified duration?

Macaulay duration is the weighted average time to receive cash flows, in years. Modified duration is Macaulay duration divided by (1 + y/m). It measures the approximate percentage price change for a unit change in yield.

When should I use effective duration instead of modified duration?

Use effective duration when cash flows change with interest rates, such as callable bonds, putable bonds and mortgage-backed securities. It reprices the bond under yield shifts using a model. Modified duration assumes cash flows are fixed.

Why does convexity matter if I already have duration?

Duration is a straight-line estimate of a curved relationship. For large yield moves the error grows. Convexity captures the curvature and improves the estimate, adding a positive amount for standard bonds.

Can convexity be negative?

Yes. Callable bonds and mortgage-backed securities can show negative convexity when yields fall and the option becomes valuable. Price gains are then limited as yields decline.