Strategic Financial Management · Equity and Bond Valuation and Evaluation of Performance

Bond Duration, Convexity and Immunization Explained

Updated 11 October 2026 · Fact-checked

Macaulay duration is the present-value-weighted average time to a bond's cash flows, in years. Modified duration is Macaulay duration divided by (1 + yield) and estimates the percentage price change for a yield change. Convexity corrects that estimate. Immunization matches asset duration to the liability horizon so interest rate moves cancel out.

Understand Bond Duration, Convexity and Immunization

A bond's price falls when market yields rise and rises when they fall. Maturity alone does not tell you how sensitive the price is, because coupons return part of your money early. Duration measures this sensitivity in one number.

Macaulay duration is the weighted average time at which you receive the bond's cash flows. Each year's weight is the present value of that cash flow divided by the bond's price. A zero-coupon bond pays everything at maturity, so its duration equals its maturity. A coupon bond has a duration shorter than its maturity. A higher coupon or a higher yield lowers duration.

Modified duration converts Macaulay duration into price sensitivity. If modified duration is 4, a 1% rise in yield cuts the price by about 4%. This is a linear estimate. The true price-yield curve is bent (convex), so the estimate is slightly off for large yield moves.

Convexity measures that bend. It is positive for ordinary bonds. For a large yield change, duration alone overstates the price fall when yields rise and understates the price gain when yields fall. Adding the convexity term fixes this. Between two bonds with the same duration, the one with higher convexity is better for the holder.

Immunization protects a portfolio from interest rate changes. You set the portfolio's duration equal to the investment horizon (the duration of the liability). Then a yield change moves the reinvestment income and the market value of the bonds in opposite directions, and the target value is protected. Because duration drifts with time and yield, you must rebalance regularly.

Key rules to remember

Macaulay duration: D = Σ [t × PV(CFt)] ÷ Σ PV(CFt) = Σ [t × PV(CFt)] ÷ P; t is the time of each cash flow in years. PV is discounted at the bond's yield to maturity. The denominator is the bond's price.
Modified duration: D* = D ÷ (1 + y ÷ m); y is the annual yield and m is the number of coupon payments per year. For annual coupons, D* = D ÷ (1 + y).
Price change using duration: ΔP ÷ P ≈ − D* × Δy; Δy in decimals, so 1% = 0.01. The sign is negative because price and yield move in opposite directions.
Convexity (annual coupons): C = [Σ t × (t + 1) × CFt ÷ (1 + y)^t] ÷ [P × (1 + y)²]; Use this form only when cash flows are annual. Check the convexity convention given in the question before using a stated value.
Price change using duration and convexity: ΔP ÷ P ≈ − D* × Δy + ½ × C × (Δy)²; The convexity term is always positive for an ordinary bond, so it adds to the price in both directions.
Zero-coupon bond duration: D = Maturity in years; Holds exactly for a bond with no interim cash flows.
Portfolio duration: Dp = Σ (wi × Di); wi is the market-value weight of each bond. Use market values, not face values.
Price value of a basis point: PVBP ≈ D* × P × 0.0001; Approximate rupee change in price for a 0.01% change in yield.
Immunization conditions: Duration of assets = Duration of liabilities (investment horizon); PV of assets ≥ PV of liabilities; Rebalance periodically as time passes and yields change.

How to solve Bond Duration, Convexity and Immunization questions

Use this sequence for any duration, convexity or immunization question.

1Write the bond details: face value, coupon rate, years to maturity, yield and coupon frequency. Convert to the correct number of periods and a periodic yield if coupons are not annual.
2List the cash flows by year. Add the face value to the final coupon.
3Discount each cash flow at the yield to maturity. Add the present values to get the bond price.
4Multiply each present value by its time t and add them up. Divide by the price to get Macaulay duration.
5Divide by (1 + y ÷ m) to get modified duration. Convert to years if you used half-year periods.
6Estimate the price change with ΔP ÷ P ≈ − D* × Δy. Add the convexity term ½ × C × (Δy)² if convexity is given or asked.
7For immunization, compute duration of assets and of the liability. Choose weights so portfolio duration equals the horizon, and check that the present value of assets covers the liability.
8State the answer with a one-line conclusion: how risky the bond is, or what the investor should do.

Quickest way: Table method with a price check

When to use it: Use it for any numerical on 3 to 5 year bonds where you must find duration within a few minutes.

Draw four columns: year t, cash flow, PV at yield, t × PV.
Fill the PV column using successive discount factors, building each factor from the previous one (divide by 1 + y again).
Add the PV column. This is the price. If the question gives a price, check that you match it.
Add the last column and divide by the price. That is Macaulay duration.
Check that it is below maturity for a coupon bond. If not, you have an arithmetic error.
Divide by (1 + y) for modified duration and multiply by Δy for the quick price change.

Common mistakes in Bond Duration, Convexity and Immunization

Dividing the t × PV total by the face value instead of the bond price.
Students treat face value as the denominator because it is the round number in the question.
Fix: The denominator is always the sum of the present values, which is the current price. Write 'Price = ΣPV' under the table.
Forgetting the face value in the final year's cash flow.
The coupon row looks complete, so the redemption amount is overlooked.
Fix: Always write the final cash flow as coupon + redemption value. Tick it off before discounting.
Using the coupon rate instead of the yield to maturity for discounting.
Both rates appear in the question, and the coupon rate is the more familiar one.
Fix: Discount at the market yield. The coupon rate only fixes the size of the cash flows.
Using Macaulay duration directly for price change.
Students remember 'duration' and skip the modified duration step.
Fix: Price change uses modified duration. Divide Macaulay duration by (1 + y ÷ m) first.
Mishandling semi-annual bonds: using annual yield with half-year periods, or reporting duration in half-years.
Periods and rates are mixed between annual and half-yearly bases.
Fix: Use y ÷ 2 per period and the number of half-year periods. Divide the duration in periods by 2 to report it in years.
Saying immunization is a one-time action, or that matching maturity is enough.
Theory is memorised as a slogan without the conditions.
Fix: State that duration of assets must equal the horizon, that present value of assets must cover the liability, and that rebalancing is needed as duration changes.

Worked examples

Example 1

A 3-year bond of face value ₹1,000 pays an annual coupon of 10%. The yield to maturity is 12%. Calculate the bond price, Macaulay duration and modified duration. Estimate the percentage fall in price if the yield rises by 1 percentage point.

Show the solution

Cash flows: Year 1 = ₹100, Year 2 = ₹100, Year 3 = ₹100 + ₹1,000 = ₹1,100.
Discount factors at 12%: Year 1 = 0.892857, Year 2 = 0.797194, Year 3 = 0.711780.
Present values: Year 1 = 100 × 0.892857 = ₹89.29. Year 2 = 100 × 0.797194 = ₹79.72. Year 3 = 1,100 × 0.711780 = ₹782.96.
Price = 89.29 + 79.72 + 782.96 = ₹951.96 (approx.).
t × PV: Year 1 = 1 × 89.29 = 89.29. Year 2 = 2 × 79.72 = 159.44. Year 3 = 3 × 782.96 = 2,348.87. Total = 2,597.60.
Macaulay duration = 2,597.60 ÷ 951.96 = 2.73 years (approx.).
Modified duration = 2.7286 ÷ 1.12 = 2.44 (approx.).
Price change for a 1% rise in yield ≈ − 2.44 × 0.01 = − 2.44%, which is about ₹23.2 on a price of ₹951.96.

Answer: Price ≈ ₹951.96. Macaulay duration ≈ 2.73 years. Modified duration ≈ 2.44. A 1 percentage point rise in yield lowers the price by about 2.44%, roughly ₹23.

Example 2

A bond is priced at ₹1,000. Its modified duration is 4.5 and its convexity is 30 (using ΔP ÷ P ≈ − D* × Δy + ½ × C × (Δy)²). Estimate the new price if the yield (a) rises by 1 percentage point and (b) falls by 1 percentage point. Explain why the two changes are not equal in size.

Show the solution

Duration effect for a 1% change = 4.5 × 0.01 = 0.045, that is 4.5%.
Convexity effect = ½ × 30 × (0.01)² = 0.5 × 30 × 0.0001 = 0.0015, that is 0.15%.
(a) Yield rises: ΔP ÷ P = − 0.045 + 0.0015 = − 0.0435, that is − 4.35%. New price = 1,000 × (1 − 0.0435) = ₹956.50.
(b) Yield falls: Δy = − 0.01, so ΔP ÷ P = + 0.045 + 0.0015 = + 0.0465, that is + 4.65%. New price = 1,000 × 1.0465 = ₹1,046.50.
The convexity term is positive in both cases. It reduces the fall and adds to the rise, so the gain from a yield fall is larger than the loss from an equal yield rise.

Answer: (a) About ₹956.50. (b) About ₹1,046.50. Positive convexity makes the price gain on a yield fall (4.65%) larger than the price loss on a yield rise (4.35%). Duration alone would show ±4.5%.

Exam tips

In MCQs, first check direction and size: price moves opposite to yield, and a coupon bond's Macaulay duration is below its maturity. This eliminates wrong options fast.
Read the yield and coupon frequency carefully. A semi-annual bond needs half-year periods, a half-year yield and a final conversion to years.
If the question gives a convexity figure, use it exactly as defined in the question. Write the formula you are applying before substituting.
For immunization questions, always conclude with the two conditions: duration of assets equals the horizon, and present value of assets is at least that of liabilities. Mention rebalancing.
In a written answer, end with a recommendation, such as which bond is riskier or how the investor should adjust the portfolio. Use a clean four-column table so marks are given for method even if the arithmetic slips.

Practice questions from Equity and Bond Valuation and Evaluation of Performance

Bond Duration, Convexity and Immunization 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, Convexity and Immunization: frequently asked questions

What is the difference between duration and convexity?

Duration is the first-order measure of how much a bond's price changes for a small yield change. Convexity is the second-order measure that captures the curvature of the price-yield relationship. You use convexity to improve the estimate when the yield change is large.

Why is modified duration used instead of Macaulay duration for price change?

Macaulay duration is a time measure in years. Dividing it by (1 + y ÷ m) gives modified duration, which is the price sensitivity to yield. That is why the price change formula uses modified duration.

Does a higher coupon increase or decrease duration?

For bonds with the same maturity and yield, a higher coupon lowers duration. More of the value comes back earlier, so the weighted average time falls. A zero-coupon bond has the highest duration for a given maturity.

How does immunization protect a portfolio?

You set the duration of the bond portfolio equal to the investment horizon. If yields rise, the bonds lose market value but reinvest coupons at higher rates, and the reverse happens when yields fall. The two effects offset, so the target value at the horizon is protected. You must rebalance as duration changes over time.