Advanced Financial Management · Security Valuation

Bond Duration and Convexity for CA Final AFM

Updated 5 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 receive the cash flows. Modified duration is Macaulay duration ÷ (1 + yield) and gives the approximate percentage price change for a 1% yield move. Convexity corrects that estimate for larger moves.

Understand Bond Duration and Convexity

A bond's price moves opposite to market yields. When yields rise, the present value of its fixed cash flows falls. The question is by how much. Duration answers that in one number.

Macaulay duration is the weighted average time, in years, until you receive the bond's cash flows. The weight of each cash flow is its present value as a share of the bond price. A bond that returns more of its value early has a shorter duration. A zero-coupon bond pays everything at maturity, so its Macaulay duration equals its maturity.

Modified duration turns that time measure into a sensitivity. It equals Macaulay duration ÷ (1 + y), where y is the yield per period. If modified duration is 4, a 1% rise in yield cuts the price by about 4%.

That estimate is a straight line. The real price-yield curve is bowed (convex). So duration overstates the fall when yields rise and understates the gain when yields fall. Convexity measures the curvature and corrects this. Higher convexity is good for the holder: smaller losses and larger gains for the same yield move.

Duration is higher for longer maturity and lower coupon. It is also higher when the yield is lower. In exams you usually compute it for a 2 to 3 year bond, then apply it to a price or portfolio-change question.

Key rules to remember

Macaulay duration: D = Σ [t × PV(CFt)] ÷ Σ PV(CFt) = Σ [t × PV(CFt)] ÷ Bond price; PV at the yield to maturity. t is in years if cash flows are annual.
Modified duration: MD = D ÷ (1 + y); For m payments a year, use MD = D ÷ (1 + y/m), with D and y on the same basis.
Price change using duration: ΔP ÷ P ≈ − MD × Δy; Δy is in decimals: 1% = 0.01. The sign is negative, so yield up means price down.
Convexity: C = Σ [CFt × t × (t + 1) ÷ (1 + y)^t] ÷ [P × (1 + y)²]; This is the standard form for annual cash flows. Check which form the question gives and use that one.
Price change using duration and convexity: ΔP ÷ P ≈ − MD × Δy + ½ × C × (Δy)²; The convexity term is always positive for a plain bond. It reduces a fall and increases a rise.
Zero-coupon bond: D = maturity in years; Coupon bonds always have Macaulay duration below maturity.

How to solve Bond Duration and Convexity questions

Use this sequence for any duration or convexity question. Build a small table. It keeps the working visible and earns step marks.

1List the cash flows: each coupon, and the redemption value added to the last coupon.
2Discount each cash flow at the given yield. Add the present values to get the bond price. If a market price is given, check that it agrees.
3Multiply each present value by its time t. Add these. Divide by the bond price to get Macaulay duration.
4Divide by (1 + y) to get modified duration. For half-yearly payments, use y/2 and convert the final answer to years.
5If convexity is asked, add a column for CF × t × (t + 1) ÷ (1 + y)^t. Add it up and divide by P × (1 + y)².
6Apply ΔP ÷ P ≈ − MD × Δy, adding ½ × C × (Δy)² if convexity is required.
7Convert the percentage into rupees and state the new approximate price. Say clearly that it is an estimate, and comment on the direction of change.

Quickest way: Table-first shortcut

When to use it: Use when the question gives a 2 to 4 period bond and asks for duration, modified duration or price change.

Draw four columns: t, cash flow, PV, and PV × t. Add a fifth column for t(t+1) × PV only if convexity is asked.
Compute PVs once with the discount factors. Reuse them in every column.
If the bond is priced at par, the PVs add up to the face value. Use that as a check.
Macaulay duration = total of the PV × t column ÷ total of the PV column. Never divide by the face value unless the price equals it.
For a zero-coupon bond, skip the table. Duration is the maturity.

Common mistakes in Bond Duration and Convexity

Dividing the total of PV × t by face value instead of the bond price.
Students are used to par bonds, where the two are equal.
Fix: Always divide by the sum of present values. That sum is the price at the given yield.
Using the undiscounted cash flows when weighting time.
The table starts with cash flows and the discounting step gets skipped.
Fix: Compute present values first. Weights come from PVs, not raw cash flows.
Plugging 1% as 1 instead of 0.01 into the price-change formula.
Mixing percentage points with decimals.
Fix: Convert Δy to a decimal. Alternatively, treat MD × 1 as a percentage change and keep the units consistent.
Forgetting the negative sign, or adding the convexity term with the wrong sign.
Only the formula is memorised, not the logic.
Fix: Duration term: price falls when yield rises. Convexity term is always a positive adjustment for a plain bond.
Not adjusting for half-yearly coupons.
Periods are counted but not converted back to years.
Fix: Use period yield y/2 for discounting and for the (1 + y/m) divisor. Divide the duration in half-years by 2 to get years.
Treating the duration-based price as exact.
The result looks precise to two decimals.
Fix: Write 'approximately'. For large yield moves, add the convexity term, and mention that duration alone is only a linear estimate.

Worked examples

Example 1

A 3-year bond has a face value of ₹1,000, pays an annual coupon of 10%, and is redeemed at par. Its yield to maturity is 10%. Calculate (a) Macaulay duration, (b) modified duration, (c) convexity, and (d) the approximate price if yield rises by 1 percentage point, using duration and convexity.

Show the solution

Cash flows: ₹100 in year 1, ₹100 in year 2, ₹1,100 in year 3.
PVs at 10%: 100 ÷ 1.10 = 90.909; 100 ÷ 1.21 = 82.645; 1,100 ÷ 1.331 = 826.446. Total = ₹1,000.00, so the price equals par.
PV × t: 90.909 × 1 = 90.909; 82.645 × 2 = 165.289; 826.446 × 3 = 2,479.339. Total = 2,735.537.
Macaulay duration = 2,735.537 ÷ 1,000 = 2.7355 years.
Modified duration = 2.7355 ÷ 1.10 = 2.4869.
Convexity: PV × t(t+1) = 90.909 × 2 = 181.818; 82.645 × 6 = 495.868; 826.446 × 12 = 9,917.355. Total = 10,595.041. Divide by (1,000 × 1.21): C = 10,595.041 ÷ 1,210 = 8.756.
Percentage price change = − 2.4869 × 0.01 + ½ × 8.756 × (0.01)² = − 2.4869% + 0.0438% = − 2.4431%.
Change in rupees = 1,000 × (− 0.024431) = − ₹24.43. New price ≈ 1,000 − 24.43 = ₹975.57.

Answer: Macaulay duration ≈ 2.74 years; modified duration ≈ 2.49; convexity ≈ 8.76. If yield rises by 1 percentage point, the price falls by about ₹24.43 to roughly ₹975.57.

Example 2

A bond fund manager holds a bond with a market price of ₹950, modified duration 4.5 and convexity 30. The manager expects yields to fall by 50 basis points. Estimate the new price (a) using duration only and (b) using duration and convexity. Which estimate is better, and why?

Show the solution

Δy = − 0.50% = − 0.005.
Duration only: ΔP ÷ P = − 4.5 × (− 0.005) = + 0.0225, which is 2.25%.
Price change = 950 × 0.0225 = ₹21.375. New price ≈ 950 + 21.375 = ₹971.38.
Convexity term: ½ × 30 × (0.005)² = ½ × 30 × 0.000025 = 0.000375, which is 0.0375%.
Total change = 2.25% + 0.0375% = 2.2875%.
Price change = 950 × 0.022875 = ₹21.73. New price ≈ 950 + 21.73 = ₹971.73.
Convexity is positive, so the straight-line duration estimate understates the gain when yields fall. The second estimate is closer to the true price.

Answer: Duration only: about ₹971.38. Duration plus convexity: about ₹971.73. The second is better because it allows for the curvature of the price-yield relationship.

Exam tips

Show the table every time, even for a short question. Step marks are given for the PV, PV × t and divisor.
Read the question for the yield basis. If coupons are half-yearly, state clearly whether your duration is in half-years or years.
In theory-plus-numerical questions, add one line of interpretation: for example, 'a 1% rise in yield reduces the price by about 2.49%'.
Check the convexity formula in the question. If a form is supplied, follow it exactly rather than using the one you memorised.
In MCQ cases, check the sign and the unit first. Many wrong options differ only in sign or in using 1 instead of 0.01.

Practice questions from Security Valuation

Bond 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.

Bond Duration and Convexity: frequently asked questions

What is the difference between Macaulay duration and modified duration?

Macaulay duration is a time measure in years: the weighted average time to receive cash flows. Modified duration is Macaulay duration divided by (1 + yield per period) and measures price sensitivity. Use Macaulay for timing and modified for estimating percentage price change.

Why does a bond's duration fall when its coupon rises?

A higher coupon returns more value earlier, so the weighted average time to receive cash flows is shorter. The maturity cash flow carries a smaller share of the total present value. That is why a zero-coupon bond has the longest duration for a given maturity.

Why do we need convexity if we already have duration?

Duration gives a straight-line estimate of price change. The actual price-yield curve is curved, so the estimate is less accurate for large yield changes. Convexity adds a correction that makes the estimate closer to the true price.

Can I use duration to find the exact new price of a bond?

No. Duration and convexity only give an approximation. The exact price comes from discounting all cash flows at the new yield. In exams, use the approximation when the question asks for duration-based change.