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FRM Part 1 · Worked study notes

Bond Pricing and Duration Convexity

Bond price is the present value of future cash flows. Modified duration estimates the price change for a small yield move; convexity adds curvature. For an option-free bond with fixed cash flows, a higher yield means a lower price.

The method at a glance

Yield up, price down

Price from 100 · modified duration 4.5 · convexity 25

9698100102104-100 bp-50 bp0 bp+50 bp+100 bpChange in yield

Solid indigo: duration + convexity. Dashed gray: duration alone. Illustrative local estimates, not observed prices or exact repricing.

At +50 bp: duration only
97.75000
With convexity
97.78125

  • 01Price

    How it works: Discount each coupon and principal payment using consistent periods and rates.

    Watch for: Using an annual yield with semiannual cash flows without converting it.

  • 02Duration

    How it works: ΔP/P ≈ -Dmod × Δy. Enter 50 basis points as 0.005.

    Watch for: Treating Macaulay duration as modified duration.

  • 03Convexity

    How it works: Add ½ × C × (Δy)² to the duration estimate.

    Watch for: Assuming a second-order estimate is exact for a large shock.

Original worked example

A 50-basis-point yield increase

A bond is priced at 100, with modified duration 4.5 and convexity 25. Assume unchanged cash flows and a parallel yield increase of 0.005.

  1. 1

    Duration contribution: -4.5 × 0.005 = -0.0225, or -2.25%.

  2. 2

    Convexity contribution: 0.5 × 25 × 0.005² = 0.0003125, or +0.03125%.

  3. 3

    Combined change: -2.21875%. Estimated price: 100 × 0.9778125.

Estimated price = 97.78125. Positive convexity reduces the estimated loss relative to duration alone.

Illustrative assumptions and calculations for learning. Not observed market data or an official exam item.

Apply the method and check its limits

Before you start: Discounting, coupon frequency and basis points. A one-basis-point yield change is 0.0001.

Calculate price and modified duration from cash flows

P = Σ CFₜ / (1 + y/m)ᵐᵗ; Dmod = DMac / (1 + y/m)

Here y is annual yield, m is payments per year and t is time in years. Macaulay duration is the present-value-weighted average payment time. For the two-year annual-pay 5% bond below, the present values are 4.8077 and 97.0784. DMac = (1 × 4.8077 + 2 × 97.0784)/101.8861 = 1.9528 years; Dmod = 1.8777. Duration is therefore derived from cash flows, not simply set equal to maturity.

Separate the yield shock from the hedge size

DV01 ≈ market value × Dmod × 0.0001

A 2 million position with modified duration 4.5 has DV01 of about 900. A 50 bp rise therefore produces a first-order loss of 45,000. With convexity 25, the second-order correction is +625, giving an estimated loss of 44,375. This assumes the same parallel shock used to calculate duration. Key-rate exposures are needed when different maturities move differently.

Price first, then measure the sensitivity

For a two-year annual-pay bond with face value 100, coupon 5 and yield 4%, price = 5/1.04 + 105/1.04² = 101.8861. The premium is consistent with the coupon exceeding the yield. A duration hedge removes only the chosen first-order exposure; a curve twist or a change in embedded-option behaviour can leave substantial residual risk.

Work through the topic

Build a bond price from its payments, derive its sensitivity, and judge when an approximation is reliable. Keep the cash-flow assumptions and yield convention consistent throughout.

Build the cash-flow table before calculating duration

Consider an option-free bond with face value 100, a 5% annual coupon and two years remaining. Immediately after a coupon date, the remaining payments are 5 after one year and 105 after two years. At a 4% annual yield, each payment has a different discount factor. Adding the discounted payments gives the full value of the remaining cash flows. The final payment contains both the coupon and principal; forgetting the coupon at maturity understates price.

The first payment contributes 4.8077 to price and the second contributes 97.0784. Their sum is 101.8861. Because the coupon rate exceeds the required yield, the bond trades above face value. That comparison is a useful reasonableness check, provided the comparison uses matching conventions and an ordinary fixed-rate bond. It does not replace discounting when payment dates, credit assumptions or embedded options differ.

Duration uses the same present values as weights. Divide each present value by the total price, multiply by its payment time and add. The weights sum to one, and Macaulay duration is 1.9528 years. It is below the two-year maturity because some value is received earlier. This weighted-time interpretation explains why a zero-coupon bond has Macaulay duration equal to its maturity, while an otherwise comparable coupon bond receives part of its value sooner.

Original annual-pay example: face 100, coupon 5, yield 4%

Year 1
Cash flow and present value: 5 / 1.04 = 4.8077
Contribution to weighted time: 1 × 4.8077 / 101.8861 = 0.0472 years
Year 2
Cash flow and present value: 105 / 1.04² = 97.0784
Contribution to weighted time: 2 × 97.0784 / 101.8861 = 1.9056 years
Total
Cash flow and present value: Price = 101.8861
Contribution to weighted time: Macaulay duration = 1.9528 years

Convert payment frequency and distinguish duration measures

Macaulay duration is a weighted payment time. Modified duration converts that time measure into a local percentage price sensitivity to the quoted yield. With annual payments, divide Macaulay duration by 1 + y. The example therefore has modified duration 1.9528/1.04 = 1.8777. The negative sign belongs in the price-change relation: a positive yield change produces a negative first-order price change for this bond.

For a nominal annual yield compounded m times per year, use y/m as the periodic rate and m payments per year. A two-year bond with a 5% nominal coupon paid semiannually pays 2.5 at each half-year and 102.5 at the final half-year. Discounting those four payments at 2% per half-year gives 101.9039. This is a different payment schedule from the annual example, so its price and sensitivity need not match exactly.

Effective duration is useful when a change in the benchmark curve can change expected cash flows. A callable bond may be redeemed earlier when rates fall. Holding its original payments fixed while changing only the discount rate misses that behavior. Reprice the instrument under up and down curve shocks using an appropriate cash-flow model. Keep the benchmark curve shock distinct from a movement in a bond-specific credit spread; a single duration number does not identify every driver.

Compare the approximation with exact repricing

Return to the annual-pay bond and raise its yield from 4% to 4.5%. Exact repricing gives 5/1.045 + 105/1.045² = 100.9363. Using modified duration alone gives 101.8861 × (1 - 1.8777 × 0.005), approximately 100.9295. The duration estimate slightly overstates the fall because it follows the tangent to a curved price-yield relationship.

For this payment schedule, annual-yield convexity is approximately 5.3728. Adding one-half times convexity times the squared yield change produces a price of 100.9364. The improvement is visible because the comparison holds the bond, cash flows, starting yield and shock constant. Do not combine a duration from one bond with a convexity from another, or confuse this derived example with the separate supplied-sensitivity example at the start of the lesson.

The remaining error comes from higher-order terms. For a larger shock, changed credit conditions, a curve twist or altered exercise behavior, full revaluation may be necessary. A hedge that offsets aggregate DV01 can still leave exposure at individual maturities. Use the risk-factor mapping lesson for those tenor exposures. A good answer reports the estimated change, the units and the conditions under which that estimate can be trusted.

Same bond, same 50 bp increase, three valuation methods

Exact cash-flow repricing
Estimated new price: 100.9363
Interpretation: Benchmark for the stated fixed payments
Modified duration only
Estimated new price: 100.9295
Interpretation: First-order approximation
Duration plus convexity
Estimated new price: 100.9364
Interpretation: Second-order approximation, still not exact

Why use 0.005 for a 50-basis-point move?

One basis point is one hundredth of a percentage point, or 0.0001 in decimal form. Fifty basis points is 0.50 percentage points, so the decimal change is 0.005. Entering 0.5 would model a 50-percentage-point jump. Write the conversion before substituting into duration or convexity formulas.

Does a fixed coupon mean a fixed market value?

No. The contractual payment can stay unchanged while investors require a different yield. The discount factors then change, moving price. Credit deterioration can also change expected recovery or required spreads. Distinguish a contractual cash flow, an expected cash flow and the price at which the claim can currently be sold.

Try a changed assumption

With price 100, modified duration 4.5 and convexity 25, what changes if yield falls 50 bp?

Show answer and reasoning

Duration contributes +2.25%; convexity still contributes +0.03125%. The estimated price is 102.28125. The convexity term is positive for either shock direction because the yield change is squared.

Original learning check. No email or account needed.

Does convexity always help?

Positive convexity raises the second-order price estimate for either direction of yield change, all else equal. Callable bonds can have negative effective convexity, and a hedge still carries curve and basis risk.

Sources and scope

Links checked 2026-10-10. GARP establishes exam scope; the technical references support the methods identified below. These are selected study notes and original examples, not complete curriculum coverage or official exam questions.

Prepared by MiloPrep editorial team. MiloPrep is independent of GARP. References do not imply endorsement or independent review of these notes.