Risk Management in Banking and Insurance · Interest Rate Risk Management

Duration and Convexity: Bond Price and Bank Equity Risk

Updated 11 October 2026 · Fact-checked

Duration measures how sensitive a bond's price is to yield changes. Macaulay duration is the present-value-weighted average time to cash flows; modified duration divides it by (1 + yield) to give percentage price change per unit yield change. Convexity corrects the estimate. Duration gap applies this to a bank's assets and liabilities to estimate the change in equity value.

Understand Duration and Convexity

A bond's price falls when market yields rise. Duration tells you by how much. It is a single number that summarises the timing of all cash flows, weighted by their present values.

Macaulay duration is the weighted average time, in years, at which you receive the bond's cash flows. The weights are the present value of each cash flow divided by the bond price. A zero-coupon bond has a duration equal to its maturity. A coupon bond has a duration shorter than its maturity. A higher coupon or a higher yield lowers duration.

Modified duration converts this into a price sensitivity: Macaulay duration ÷ (1 + yield). If modified duration is 4, a 1% rise in yield cuts the price by about 4%. This is a linear estimate and works well only for small yield changes.

The true price-yield curve is bent, not straight. Convexity measures that bend. For a bond with normal cash flows, convexity is positive, so duration alone overstates the fall when yields rise and understates the gain when yields fall. Adding a convexity term gives a closer estimate for larger moves.

Banks hold assets and liabilities with different durations. The duration gap combines them into one figure. A positive gap means asset values fall by more than liability values when rates rise, so equity value falls. This is the economic value view of interest rate risk.

Key rules to remember

Macaulay duration: D = Σ [t × PV(CFt)] ÷ P, where P = Σ PV(CFt); t is time in years. PV is discounted at the yield to maturity. Result is in years.
Modified duration: D_mod = D ÷ (1 + y/m); m is the number of coupon payments per year. For annual coupons, divide by (1 + y).
Price change using duration: ΔP ÷ P ≈ − D_mod × Δy; Use Δy as a decimal (1% = 0.01). The sign is negative: yield up, price down.
Convexity (annual cash flows): C = [Σ t(t + 1) × PV(CFt)] ÷ [P × (1 + y)²]; Measured in years squared. Use the version given in the question if it differs for non-annual periods.
Price change with convexity: ΔP ÷ P ≈ − D_mod × Δy + ½ × C × (Δy)²; The convexity term is positive for both rises and falls in yield.
Duration gap: DGAP = D_A − (L ÷ A) × D_L; A is the market value of assets, L of liabilities. D_A and D_L are the weighted durations.
Change in equity value: ΔE ≈ − DGAP × A × Δy ÷ (1 + y); Positive DGAP: equity falls when rates rise. Negative DGAP: equity rises when rates rise.

How to solve Duration and Convexity questions

Use this order for any question on duration, convexity or duration gap. It keeps the workings clear and easy to mark.

1Identify what is asked: Macaulay duration, modified duration, convexity, price change, or change in bank equity.
2List the cash flows by year, including the face value in the final year. Note the coupon frequency and the yield.
3Discount each cash flow at the yield and add them to get the price P. Check it against any price given.
4Compute Σ t × PV for duration, and Σ t(t + 1) × PV for convexity, in a neat column table.
5Divide to get Macaulay duration, then modified duration by dividing by (1 + y/m). Compute convexity if required.
6Apply the price change formula. Use Δy as a decimal. Add the convexity term only if asked or if the yield change is large.
7For a bank, find the value-weighted D_A and D_L, then DGAP, then ΔE. Express ΔE in rupees and, if useful, as a percentage of equity.
8State the conclusion: direction of change, size, and what the bank should do, for example shorten asset duration or hedge.

Quickest way: Shortcut for price and equity estimates

When to use it: Use this when duration, convexity or the gap is already given and you only need the change in value. It saves the full cash flow table.

Write down D_mod (or Macaulay duration and y) and Δy as a decimal.
Duration effect: − D_mod × Δy. Convert to rupees by multiplying with the price.
Convexity effect: ½ × C × Δy². Add it to the duration effect.
For banks: DGAP first, then ΔE = − DGAP × A × Δy ÷ (1 + y).
Sense-check the sign: rates up with positive DGAP must reduce equity.

Common mistakes in Duration and Convexity

Using modified duration where Macaulay duration is needed, or the reverse.
The two names sound alike and both are called duration in the question.
Fix: Remember that Macaulay is in years and measures time. Modified is the price sensitivity and equals Macaulay ÷ (1 + y/m). Use modified for price change.
Putting Δy as 1 instead of 0.01.
Students keep the yield change in percentage points when multiplying.
Fix: Convert every yield change to a decimal before substituting. Check that the answer is a sensible percentage.
Forgetting the face value in the final year's cash flow.
Attention goes to the coupon column only.
Fix: In the last year, the cash flow is coupon plus redemption value. Check that the PV total equals the price.
Dropping the minus sign or applying duration to the wrong direction.
Students focus on the arithmetic and not the logic.
Fix: State the direction in words first: yield up means price down. For the gap, positive DGAP and rising rates mean falling equity.
Leaving out the (L ÷ A) weight in the duration gap.
Students subtract D_L from D_A directly.
Fix: Liabilities fund only part of the assets. Always use D_A − (L ÷ A) × D_L, with market values.
Treating convexity as always lowering the price.
The convexity term is added with a plus sign, but students confuse it with the duration term.
Fix: For a normal bond the convexity adjustment is positive in both directions. It reduces the fall or increases the rise compared with the duration estimate.

Worked examples

Example 1

A 3-year bond of face value ₹1,000 pays an annual coupon of 10% and is priced to yield 10%. Calculate its Macaulay duration, modified duration and convexity. Then estimate the price after the yield rises by 1 percentage point, with and without the convexity adjustment.

Show the solution

Cash flows: ₹100 in year 1, ₹100 in year 2, ₹1,100 in year 3. Discount at 10%.
PV: 100 ÷ 1.10 = 90.909; 100 ÷ 1.21 = 82.645; 1,100 ÷ 1.331 = 826.446. Price P = ₹1,000, which is par as expected.
Σ t × PV = 1 × 90.909 + 2 × 82.645 + 3 × 826.446 = 90.909 + 165.289 + 2,479.339 = 2,735.537.
Macaulay duration = 2,735.537 ÷ 1,000 = 2.7355 years.
Modified duration = 2.7355 ÷ 1.10 = 2.4869.
Σ t(t + 1) × PV = 2 × 90.909 + 6 × 82.645 + 12 × 826.446 = 181.818 + 495.868 + 9,917.355 = 10,595.041.
Convexity = 10,595.041 ÷ (1,000 × 1.21) = 8.756.
Duration only: ΔP ÷ P = − 2.4869 × 0.01 = − 2.487%. Fall = ₹24.87. Price ≈ ₹975.13.
Convexity term: ½ × 8.756 × 0.0001 = 0.000438 = 0.044%. Total change = − 2.487% + 0.044% = − 2.443%. Fall = ₹24.43. Price ≈ ₹975.57.

Answer: Macaulay duration is about 2.74 years, modified duration about 2.49, and convexity about 8.76. The estimated price is ₹975.13 with duration only and ₹975.57 with the convexity adjustment.

Example 2

A bank has assets of ₹10,000 crore with a weighted duration of 4.0 years, and liabilities of ₹9,200 crore with a weighted duration of 2.5 years. The market yield is 10%. Compute the duration gap and estimate the change in the bank's equity value if yields rise by 1 percentage point. Comment on the result.

Show the solution

Equity = 10,000 − 9,200 = ₹800 crore.
L ÷ A = 9,200 ÷ 10,000 = 0.92.
DGAP = 4.0 − 0.92 × 2.5 = 4.0 − 2.30 = 1.70 years.
ΔE ≈ − DGAP × A × Δy ÷ (1 + y) = − 1.70 × 10,000 × 0.01 ÷ 1.10.
ΔE = − 170 ÷ 1.10 = − ₹154.55 crore.
New equity ≈ 800 − 154.55 = ₹645.45 crore. The fall is 154.55 ÷ 800 = 19.3% of equity.

Answer: The duration gap is 1.70 years. Equity value falls by about ₹154.55 crore, roughly 19.3% of equity. The bank is exposed to rising rates because asset duration is longer than liability duration. It can shorten asset duration, lengthen liability duration, or hedge with swaps or other derivatives.

Exam tips

In MCQs, check the formula's condition first. Questions often ask for modified duration when Macaulay duration is given. One division by (1 + y) is the whole step.
Draw the PV table with columns t, cash flow, PV, t × PV and t(t + 1) × PV. It gets method marks even if you slip on one figure.
For a duration gap question, always give the sign and a recommendation. Say which side to adjust or hedge.
Keep two or three decimals in duration and convexity until the last step. Early rounding changes the answer in the option list.
In descriptive answers, also state the limits: duration assumes a parallel yield shift and small changes, and convexity improves but does not remove the error.

Practice questions from Interest Rate Risk Management

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 Macaulay duration and modified duration?

Macaulay duration is the weighted average time to receive a bond's cash flows, measured in years. Modified duration equals Macaulay duration divided by (1 + y/m). It tells you the approximate percentage price change for a one-unit change in yield.

Why is convexity added to the duration estimate?

Duration gives a straight-line estimate, but the real price-yield relationship is curved. For a normal bond the curve lies above the line, so duration overstates losses and understates gains. The convexity term corrects this for larger yield changes.

What does a positive duration gap mean for a bank?

It means the assets are more rate-sensitive in value than the liabilities. When interest rates rise, asset values fall by more than liability values and equity value declines. When rates fall, equity value rises.

Does a bond with a higher coupon have a higher or lower duration?

A higher coupon gives a lower duration, other things being equal. More of the value comes earlier, so the weighted average time to cash flows is shorter. A zero-coupon bond has the highest duration for its maturity.