FRM Exam Part I · Applying Duration, Convexity, and DV01

Price-Yield Relationship and Bond Pricing Basics

Q: Why is the bond price-yield curve convex?

Discount factors 1 ÷ (1 + y)^t are convex in y, and a bond price is a sum of them with positive weights. So the sum is convex too. The slope becomes less steep as yield rises.

Q: Why does a lower coupon bond have more price sensitivity?

Less of its value arrives early, so more of the price depends on distant, heavily discounted cash flows. A change in yield affects distant cash flows more. Its duration is therefore longer.

Q: Is the convexity adjustment always positive?

For a standard bond with fixed cash flows, yes, because it uses (Δy)² and convexity is positive. Bonds with embedded options, such as callable bonds or mortgage securities, can show negative convexity.

Q: How do I price a bond on the FRM calculator?

Enter N as the number of periods, I/Y as the per-period yield in percent, PMT as the per-period coupon and FV as face value. Then compute PV. Ignore the negative sign.

Updated 11 October 2026 · Fact-checked

A bond's price is the present value of its coupons and principal, discounted at the yield. When the yield rises, the price falls, and the reverse. The price-yield curve is convex, so a straight-line estimate from duration misses part of the move. You then add a convexity correction.

Understand Price-Yield Relationship and Bond Pricing Basics

A bond is a stream of fixed cash flows: coupons, then principal at maturity. Its price is the present value of those cash flows. You discount each one at the bond's yield to maturity (YTM). Because a higher discount rate shrinks every present value, price and yield move in opposite directions.

The relationship is not a straight line. It is a curve that bends upward, called convex. Why? Discount factors are 1 ÷ (1 + y)^t. This function flattens as y rises. So a fall in yield raises the price by more than an equal rise in yield lowers it. For the same size of yield move, the gain is larger than the loss.

Duration is the slope of the curve at the current yield. It gives a straight-line (tangent) estimate of the price change. The tangent lies below the true curve everywhere except at the starting point. So duration alone underestimates the price after a fall in yield and overestimates the drop after a rise. The error grows with the size of the yield move. Convexity measures the curvature and corrects this.

Three features drive sensitivity. Maturity: longer maturity means more sensitivity, other things equal. Coupon: lower coupon means more sensitivity, because more of the value arrives late. A zero-coupon bond is the most sensitive for its maturity. Yield level: at lower yields, the bond is more sensitive and more convex.

A bond priced at par has coupon rate equal to YTM. A bond with coupon above YTM trades at a premium. A bond with coupon below YTM trades at a discount. As maturity approaches, price pulls toward par.

Key formulas to remember

Bond price (annual coupons): P = Σ C ÷ (1 + y)^t + F ÷ (1 + y)^T; C is the coupon, F the face value, T the years to maturity, y the annual YTM. For semiannual coupons use C ÷ 2, y ÷ 2 and 2T periods.
Zero-coupon bond price: P = F ÷ (1 + y)^T; Single cash flow, so the calculation is one step.
Premium, par, discount: Coupon rate > y → P > F; coupon rate = y → P = F; coupon rate < y → P < F; Quick check on whether your answer is sensible.
Duration approximation: ΔP ÷ P ≈ −D_mod × Δy; D_mod is modified duration. First-order, linear estimate.
Duration plus convexity approximation: ΔP ÷ P ≈ −D_mod × Δy + ½ × C × (Δy)²; C is convexity. The convexity term is positive for option-free bonds, whether yields rise or fall.

How to solve Price-Yield Relationship and Bond Pricing Basics questions

Use this order for any price-yield or bond pricing question.

1Identify the cash flows: coupon per period, face value, number of periods. Match the coupon frequency to the compounding of the yield.
2Convert the yield and coupon to a per-period basis. For semiannual: divide annual rates by 2 and double the years.
3Discount each cash flow, or use the financial calculator: N, I/Y, PMT, FV, then compute PV.
4Check the sign and size against the par rule: premium, par or discount.
5If the question asks for a price change, compute the duration estimate: −D_mod × Δy × P.
6Add the convexity term ½ × C × (Δy)² × P if convexity is given or the move is large.
7Compare with the full repricing if needed. The difference is the approximation error, and it is positive-signed in favour of the true price for option-free bonds.

Quickest way: Calculator pricing and direction checks

When to use it: When the question gives coupon, maturity and yield and asks for price, or asks which bond moves more.

Enter N (periods), I/Y (per-period yield in %), PMT (per-period coupon), FV (face value), then CPT PV. The result is negative; ignore the sign.
Rank sensitivity by rule: longer maturity and lower coupon mean larger price changes for a given yield move.
Eliminate options that show price rising with yield, or a convexity correction with the wrong sign.
For a price change, compute only the duration term first. If the options differ by more than the convexity term, you are done.

Common mistakes in Price-Yield Relationship and Bond Pricing Basics

Using the annual yield with semiannual coupons without adjusting.
Questions quote yields as annual, and you plug them straight in.
Fix: Halve the yield and the coupon, and double N, whenever coupons are paid twice a year.
Thinking the price-yield curve is a straight line.
Duration is taught first and looks like a slope rule.
Fix: Remember duration is only the tangent. The real curve is convex, so the tangent underestimates the price.
Subtracting the convexity term for a yield rise.
You assume convexity should hurt when yields go up.
Fix: The term ½ × C × (Δy)² is positive in both directions because Δy is squared. Convexity always helps an option-free bond.
Assuming higher coupon means more price sensitivity.
Larger payments feel like greater exposure.
Fix: A higher coupon returns cash earlier, which lowers duration. Lower coupon, same maturity, means higher sensitivity.
Entering the yield as a decimal in I/Y on the calculator.
Formula work uses 0.05, so you carry it across.
Fix: Enter 5 for 5%. Check that the price is reasonable using the par rule.

Worked examples

Example 1

A 3-year bond pays an annual coupon of 6% on a face value of USD 1,000. The YTM is 8% (annual compounding). What is its price?

Show the solution

Coupon = 6% × 1,000 = 60 per year. Face value = 1,000.
PV of year 1 coupon = 60 ÷ 1.08 = 55.556.
PV of year 2 coupon = 60 ÷ 1.08² = 60 ÷ 1.1664 = 51.440.
PV of year 3 coupon plus principal = 1,060 ÷ 1.08³ = 1,060 ÷ 1.259712 = 841.447.
Sum = 55.556 + 51.440 + 841.447 = 948.44.
Check: coupon 6% is below yield 8%, so the price should be below 1,000. It is.

Answer: Price ≈ USD 948.44, a discount to par.

Example 2

A bond priced at USD 100 has modified duration 7.0 and convexity 60. Yields fall by 100 basis points. Estimate the new price using duration and convexity.

Show the solution

Δy = −0.01.
Duration term: −7.0 × (−0.01) = +0.07, so +7.00% of price.
Convexity term: ½ × 60 × (0.01)² = 30 × 0.0001 = 0.003, so +0.30%.
Total change = 7.00% + 0.30% = 7.30%.
New price = 100 × 1.0730 = 107.30.
Duration alone would give 107.00, which is below the better estimate, as expected for a convex curve.

Answer: Estimated new price ≈ 107.30.

Exam tips

Questions often ask which bond is most sensitive. Use the rule: longest maturity, lowest coupon, lowest yield.
Check the direction of the convexity adjustment. For option-free bonds it is always added.
Read the compounding frequency in the question stem before touching the calculator.
If a numeric answer is near par, test it with the premium and discount rule to catch input errors.
Conceptual items may ask why duration under- or overstates the price. The answer is the curvature of the price-yield curve.

Practice questions from Applying Duration, Convexity, and DV01