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NISM-Series-XXI-A: Portfolio Management Services (PMS) Distributors · Investing in Fixed Income Securities (NISM XXI-A)

Interest Rate Risk, Duration and Convexity Explained

Updated 11 October 2026 · Fact-checked

Interest rate risk is the risk that bond prices fall when yields rise. Modified duration estimates the percentage price change for a 1% (100 bps) yield change: ΔP% ≈ −Modified Duration × Δy. Convexity corrects this estimate, because the price-yield curve bends. Higher duration means higher sensitivity.

Understand Interest Rate Risk, Duration and Convexity

A bond pays fixed coupons. When market yields rise, new bonds pay more, so your old bond is worth less. When yields fall, your old bond is worth more. This is the inverse price-yield relationship. The risk that prices move because yields move is interest rate risk.

Not all bonds react equally. A long-maturity bond falls more than a short one for the same yield rise. A low-coupon bond falls more than a high-coupon bond of the same maturity. Duration puts a number on this sensitivity.

Macaulay duration is the weighted average time, in years, to receive the bond's cash flows, with weights equal to the present value of each cash flow. A zero-coupon bond's Macaulay duration equals its maturity. A coupon bond's is shorter than its maturity.

Modified duration converts this into price sensitivity. A modified duration of 5 means the price changes by about 5% for a 1% change in yield, in the opposite direction. It is a linear estimate, so it works well for small yield changes.

The price-yield relationship is a curve, not a straight line. Convexity measures this curvature. For a plain bond, convexity is positive. Duration alone understates the price gain when yields fall and overstates the price loss when yields rise. Adding the convexity adjustment gives a better estimate, especially for large yield moves. All else equal, higher convexity is good for the bondholder.

Key formulas to remember

Modified duration
Modified Duration = Macaulay Duration ÷ (1 + y/n)
y is the yield to maturity as a decimal; n is the number of coupon payments per year. For annual coupons, divide by (1 + y).
Price change using duration
ΔP/P ≈ −Modified Duration × Δy
The minus sign shows the inverse relationship. Use Δy as a decimal, or in % to get the answer in %.
Convexity adjustment
ΔP/P ≈ −MD × Δy + ½ × Convexity × (Δy)²
The convexity term is always positive for a plain bond, so it adds to the price change in both directions.
Duration of a zero-coupon bond
Macaulay Duration = Time to maturity
For a coupon-paying bond, Macaulay duration is less than maturity.
Direction rule
Yield ↑ → Price ↓; Yield ↓ → Price ↑
Holds for plain fixed-rate bonds.

How to solve Interest Rate Risk, Duration and Convexity questions

Use this method for any question on interest rate risk, duration or convexity.

  1. 1Identify what is asked: direction of price change, size of change, or comparison of bonds.
  2. 2For direction, apply the inverse rule: yield up means price down.
  3. 3For size, write down the modified duration and the yield change in the same units (% or decimal).
  4. 4Compute ΔP% = −MD × Δy. A 1% rise with MD of 6 gives −6%.
  5. 5If the question gives convexity, add ½ × Convexity × (Δy)² using Δy as a decimal.
  6. 6Apply the percentage change to the starting price to get the new price in rupees.
  7. 7For comparisons, remember: longer maturity and lower coupon mean higher duration and more risk.
  8. 8Check that your sign and the size look sensible before choosing an option.

Quickest way: Sign, then multiply

When to use it: Use for direct MCQs asking for approximate price change or which bond is riskier.

  1. Decide the sign first: yield up gives a fall, yield down gives a rise.
  2. Multiply modified duration by the yield change in percentage points.
  3. Eliminate options with the wrong sign at once.
  4. For ranking bonds, pick the one with longer maturity and lower coupon as most sensitive.
  5. Use convexity only if the question gives it or asks about large yield moves.

Common mistakes in Interest Rate Risk, Duration and Convexity

  • Using Macaulay duration directly as the percentage price change.

    Both are called duration and are measured in similar numbers.

    Fix: Use modified duration for price change. Divide Macaulay duration by (1 + y/n) first.

  • Forgetting the negative sign.

    Students focus on the size and ignore the direction.

    Fix: Write the sign before calculating. Yield up means price down.

  • Mixing basis points and percentages.

    A change of 50 bps is read as 50%.

    Fix: Convert first: 100 bps = 1%. So 50 bps = 0.5%.

  • Thinking a high-coupon bond has higher duration.

    Students link a high coupon with high risk.

    Fix: Higher coupons return cash sooner, so duration is lower. Lower coupon means higher duration.

  • Believing duration is exact for large yield changes.

    The formula looks precise.

    Fix: Duration is a linear approximation. For large moves, convexity improves the estimate.

  • Treating convexity as a risk to avoid.

    Students think any extra measure means extra risk.

    Fix: For plain bonds, positive convexity helps the holder: gains from falling yields exceed losses from rising yields.

Worked examples

Example 1

A bond is priced at ₹1,000 with a modified duration of 4.5. If yields rise by 50 bps, what is the approximate new price using duration only?

Show the solution
  1. Convert the yield change: 50 bps = 0.5%.
  2. Percentage price change = −4.5 × 0.5% = −2.25%.
  3. Rupee change = 2.25% of ₹1,000 = ₹22.50 fall.
  4. New price = ₹1,000 − ₹22.50 = ₹977.50.

Answer: About ₹977.50

Example 2

A bond has a Macaulay duration of 5.25 years, annual coupons and a yield to maturity of 5%. What is its modified duration, and what is the approximate percentage price change if yield falls by 1%?

Show the solution
  1. Modified duration = 5.25 ÷ (1 + 0.05) = 5.25 ÷ 1.05.
  2. 5.25 ÷ 1.05 = 5.
  3. Price change = −5 × (−1%) = +5%.
  4. The price rises by about 5%, because yield fell.

Answer: Modified duration is 5; price rises by about 5%

Exam tips

  • Expect numeric questions that need only one multiplication. Learn the formula cold and save time.
  • Watch the units. Questions often give yield changes in basis points.
  • Look for ranking questions: longer maturity and lower coupon mean higher duration.
  • Remember that a zero-coupon bond's Macaulay duration equals its maturity.
  • Negative marking applies, so skip a calculation only if you cannot fix the sign and the units.

Practice questions from Investing in Fixed Income Securities (NISM XXI-A)

Interest Rate Risk, 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.

Interest Rate Risk, Duration and Convexity: frequently asked questions

What is modified duration in simple words?

It is the approximate percentage change in a bond's price for a 1% change in yield. A modified duration of 7 means the price moves about 7% the opposite way to yield.

How is Macaulay duration different from 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 + y/n) and measures price sensitivity.

Why does convexity matter?

Duration gives a straight-line estimate, but the price-yield curve bends. Convexity corrects the error, especially for large yield changes. Positive convexity benefits the bondholder.

Which bond has the highest interest rate risk?

Other things equal, the bond with the longest maturity and the lowest coupon has the highest duration, so it has the highest interest rate risk.