NISM-Series-X-A: Investment Adviser (Level 1) · Investing in Fixed Income Securities

Interest Rate Risk, Duration and Convexity for NISM Investment Adviser Level 1

Q: How do I calculate the duration of a bond?

Find the present value of each cash flow at the bond's yield. Multiply each by its time in years and add them up. Divide by the bond's price. That gives Macaulay duration.

Q: What is convexity in bonds?

Convexity measures the curvature of the price-yield relationship. It corrects the duration estimate when yields change a lot. For a plain bond it is positive, so prices rise more for a fall in yield than they fall for an equal rise.

Q: Which bond has the highest interest rate risk?

Other things equal, the bond with the highest duration. That is usually the one with the longest maturity and the lowest coupon, such as a long-dated zero-coupon bond.

Updated 11 October 2026 · Fact-checked

Interest rate risk is the risk that bond prices fall when yields rise. Macaulay duration is the weighted average time, in years, to receive a bond's cash flows. Modified duration is Macaulay duration ÷ (1 + y), and it gives the approximate % price change for a 1% yield change. Convexity corrects that estimate for large moves.

Understand Interest Rate Risk, Duration and Convexity

A bond pays fixed cash flows. When market yields rise, those fixed payments are discounted at a higher rate, so the bond's price falls. When yields fall, the price rises. This inverse link is interest rate risk. It is the main risk in government securities and high-quality bonds.

Not every bond reacts equally. A long-maturity bond moves more than a short one. A low-coupon bond moves more than a high-coupon bond. Duration packs these effects into one number.

Macaulay duration is the weighted average time to receive the bond's cash flows, using the present value of each cash flow as its weight. It is measured in years. For a zero-coupon bond it equals the maturity. For a coupon-paying bond it is always less than the maturity.

Modified duration converts Macaulay duration into a price sensitivity: Macaulay duration ÷ (1 + yield per period). A modified duration of 4 means the price falls about 4% if yield rises by 1%, and rises about 4% if yield falls by 1%.

Duration assumes a straight-line link between price and yield. The real price-yield curve is bent. This bend is convexity. For a normal bond with positive convexity, the price rises more when yields fall than it falls when yields rise by the same amount. Convexity matters most for large yield changes, and duration alone then understates the price gain and overstates the price loss.

Key formulas to remember

Macaulay duration: D = Σ [t × PV(CFt)] ÷ Price; t is the time of each cash flow in years. PV(CFt) is the present value of that cash flow at the bond's yield. Price is the sum of all PVs.
Modified duration: Modified duration = Macaulay duration ÷ (1 + y ÷ n); y is the annual yield and n is the number of coupon payments per year. For annual coupons, divide by (1 + y).
Approximate price change using duration: % change in price ≈ − Modified duration × Δy; Δy is the change in yield in percentage points. The sign is negative because price and yield move in opposite directions.
Price change with convexity adjustment: % change in price ≈ − Modified duration × Δy + ½ × Convexity × (Δy)²; Use Δy as a decimal here (0.01 for 1%). The convexity term is always positive for a plain bond, so it adds to the estimate.
Price value of a basis point (PVBP): PVBP ≈ Modified duration × Price × 0.0001; Approximate rupee change in price for a 1 basis point (0.01%) change in yield.
Duration rules of thumb: Zero-coupon bond: D = maturity. Coupon bond: D < maturity.; Duration rises with longer maturity. It falls with a higher coupon and with a higher yield, other things equal.

How to solve Interest Rate Risk, Duration and Convexity questions

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

1Read what is given: Macaulay or modified duration, yield, coupon, maturity, price and the yield change.
2Check the question type. Is it a concept question (which bond is riskier) or a calculation (price change, duration)?
3If you are given Macaulay duration and need price sensitivity, first convert: modified duration = Macaulay duration ÷ (1 + y ÷ n).
4Convert the yield change to percentage points. 50 basis points = 0.50%.
5Apply % change in price ≈ − modified duration × Δy. Get the sign right: yield up means price down.
6If the question mentions convexity or a large yield change, add the convexity term ½ × C × (Δy)² with Δy as a decimal.
7If a rupee change is asked, multiply the % change by the bond's price.
8Check the answer: the direction should be opposite to the yield move, and the size should be reasonable.

Quickest way: Duration shortcut for MCQs

When to use it: Use when the question gives duration and a yield change and asks for the approximate price move, or asks which bond is most sensitive.

Pick the modified duration. If only Macaulay is given, divide by (1 + y).
Multiply by the yield change in percentage points.
Flip the sign against the yield move.
For comparison questions, rank by duration without calculating: longer maturity and lower coupon means higher duration and higher risk.
If two options differ only by the convexity effect, remember that a positive convexity gives a slightly better outcome in both directions.

Common mistakes in Interest Rate Risk, Duration and Convexity

Using Macaulay duration directly to estimate price change.
Both are called duration and are close in value, so students skip the conversion.
Fix: Price sensitivity uses modified duration. Divide Macaulay by (1 + y ÷ n) first.
Getting the sign wrong and saying the price rises when yields rise.
Students focus on the 'rise' and forget the inverse relationship.
Fix: Write the minus sign in the formula every time. Yield up means price down.
Treating a 50 basis point change as 50%, or as 5%.
Confusion between basis points and percentage points.
Fix: 100 basis points = 1%. So 50 basis points = 0.5%.
Saying a coupon bond's duration equals its maturity.
This is true only for zero-coupon bonds.
Fix: For a bond with coupons, duration is less than maturity because some cash comes earlier.
Thinking a higher coupon means higher interest rate risk.
Students link higher coupon with a bigger bond payoff.
Fix: A higher coupon returns cash earlier, so duration falls and price sensitivity falls, other things equal.
Believing duration gives the exact price change.
The formula looks precise.
Fix: Duration is a linear estimate. It is accurate only for small yield changes. Convexity improves it for larger changes.

Worked examples

Example 1

A bond has a Macaulay duration of 5.2 years and a yield to maturity of 8% a year, with annual coupons. Its price is ₹1,000. If the yield rises by 50 basis points, what is the approximate change in price using duration alone?

Show the solution

Convert to modified duration: 5.2 ÷ 1.08 = 4.8148.
Yield change: 50 basis points = 0.50%.
% change in price ≈ − 4.8148 × 0.50 = − 2.407%.
Rupee change ≈ 2.407% × ₹1,000 = ₹24.07 fall.

Answer: The price falls by about 2.41%, or roughly ₹24.07, to about ₹975.93.

Example 2

A 2-year bond with face value ₹1,000 pays an annual coupon of 10% and has a yield to maturity of 10%. Calculate its Macaulay duration and modified duration.

Show the solution

The bond yields 10% and the coupon is 10%, so it trades at par. Price = ₹1,000.
PV of year 1 cash flow: 100 ÷ 1.10 = ₹90.909.
PV of year 2 cash flow (coupon plus principal): 1,100 ÷ 1.21 = ₹909.091.
Check: 90.909 + 909.091 = ₹1,000.
Weighted sum: (1 × 90.909) + (2 × 909.091) = 90.909 + 1,818.182 = 1,909.091.
Macaulay duration = 1,909.091 ÷ 1,000 = 1.909 years.
Modified duration = 1.909 ÷ 1.10 = 1.7355.

Answer: Macaulay duration is about 1.91 years, which is less than the 2-year maturity. Modified duration is about 1.74.

Exam tips

Know the direction rules cold: yield up, price down. Longer maturity and lower coupon mean higher duration and higher risk.
Check whether the question gives Macaulay or modified duration. Examiners often give Macaulay to test whether you convert.
Read the yield change carefully. Basis points and percentage points are a common trap in options.
For conceptual convexity questions, remember that positive convexity benefits the bondholder: gains from falling yields are larger than losses from equal rising yields.
With negative marking, skip a calculation question only if you cannot set it up. Most are one-step once you pick modified duration.

Practice questions from Investing in Fixed Income Securities

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 the difference between Macaulay duration and modified duration?

Macaulay duration is the weighted average time, in years, to receive a bond's cash flows. Modified duration is Macaulay duration divided by (1 + yield per period). Modified duration is the one used to estimate the percentage price change for a yield change.

How do I calculate the duration of a bond?

Find the present value of each cash flow at the bond's yield. Multiply each by its time in years and add them up. Divide by the bond's price. That gives Macaulay duration.

What is convexity in bonds?

Convexity measures the curvature of the price-yield relationship. It corrects the duration estimate when yields change a lot. For a plain bond it is positive, so prices rise more for a fall in yield than they fall for an equal rise.

Which bond has the highest interest rate risk?

Other things equal, the bond with the highest duration. That is usually the one with the longest maturity and the lowest coupon, such as a long-dated zero-coupon bond.