Factors That Affect Bond Duration: Coupon, Maturity and Yield

Understand Duration Properties and Determinants

Key formulas to remember

Modified duration

ModDur = MacDur ÷ (1 + y/m)

y is the annual yield-to-maturity and m is the number of compounding periods per year; Macaulay duration is stated in years.

Zero-coupon bond

MacDur = time to maturity (in years)

Modified duration = maturity ÷ (1 + y/m). It is lower than maturity because of the divisor.

Duration between coupon dates (approximation)

MacDur (between dates) ≈ MacDur at last coupon date − time elapsed (in years)

A simplified rule of thumb, not a standard CFA formula. It assumes no change in yield and shows the downward slope of the sawtooth pattern. Duration jumps up right after a coupon is paid.

Floating-rate note

MacDur ≈ time until next reset

This holds when the spread over the reference rate matches the required spread, so the price stays near par at resets.

Approximate price change

%ΔPrice ≈ −ModDur × ΔYield

A first-order estimate. It ignores convexity, so it is more accurate for small yield changes.

Direction rules

Duration ↑ when: maturity ↑, coupon ↓, yield ↓

Other factors held constant. Maturity is a general rule: for par or premium coupon bonds duration rises at a decreasing rate toward the perpetuity duration (1 + y) ÷ y (annual coupons, annual yield y), staying below it. For deep-discount coupon bonds it may rise, or decline slightly at very long maturities as it converges to that perpetuity level, but it stays below maturity. A callable or putable bond's effective duration is less than or equal to that of an otherwise identical option-free bond, with a material reduction when the option is near or in the money.

How to solve Duration Properties and Determinants questions

1. 1Identify what is asked: Macaulay, modified or effective duration, and whether it is a comparison or a number.
2. 2List the features: coupon rate, maturity, yield, bond type (zero-coupon, FRN, callable, putable) and the date relative to coupon dates.
3. 3For comparisons, change one factor at a time and apply the rule: longer maturity generally raises duration, higher coupon or higher yield lowers it.
4. 4For a zero-coupon bond, set Macaulay duration equal to maturity. For an FRN, set it near the time to the next reset.
5. 5For embedded options, say whether the option is likely to be exercised. A callable bond at low yields or a putable bond at high yields has a materially shorter effective duration. If the option is far out of the money, expect duration close to the option-free bond. It is never above it.
6. 6For a number, convert with ModDur = MacDur ÷ (1 + y/m), using the periodic yield y/m, not the annual yield.
7. 7For a date between coupons, expect duration to be lower than at the last coupon date by roughly the time elapsed, if yield is unchanged.
8. 8Check your answer for sense: modified duration is below Macaulay duration, and a coupon bond's Macaulay duration is below its maturity, even for a deep-discount bond.

Quickest way: Rank by the three levers, then do one division

Common mistakes in Duration Properties and Determinants

• Dividing Macaulay duration by (1 + annual yield) when the bond pays semiannually.

  The formula is memorised as (1 + y) and the compounding frequency is ignored.

  Fix: Divide by (1 + y/m). For a 6% yield with semiannual periods, use 1.03, not 1.06. A wrong option built from the annual yield is a common trap.

• Saying a higher coupon raises duration because the bond pays more.

  Confusing size of payments with timing of payments.

  Fix: Duration is a weighted average time. Larger early coupons bring the average time forward, so duration falls.

• Treating a coupon bond's duration as equal to its maturity.

  The zero-coupon result is overgeneralised.

  Fix: Only a zero-coupon bond has Macaulay duration exactly equal to maturity. A coupon bond's Macaulay duration is below its maturity at a positive yield, even for a deep-discount bond.

• Assuming duration always rises with maturity.

  The simple rule is applied to every bond without checking the price level.

  Fix: For par or premium coupon bonds, duration rises with maturity at a decreasing rate toward the perpetuity duration (1 + y) ÷ y, not toward maturity. For deep-discount coupon bonds, duration may rise or decline slightly at very long maturities as it converges to that level, but it remains below maturity.

• Assuming a floating-rate note has duration equal to its maturity.

  Looking at the final maturity instead of the reset schedule.

  Fix: The coupon resets to market rates, so price stays near par. Duration is about the time to the next reset.

• Assuming an embedded option always cuts effective duration by a large amount.

  The option effect is remembered as unconditional.

  Fix: A callable or putable bond's effective duration is less than or equal to the option-free bond's. The reduction is material when the option is near or in the money. If it is far out of the money, effective duration is similar to the option-free bond's.

• Assuming a callable bond's duration keeps rising as yields fall, as for an option-free bond.

  Ignoring that the issuer's call option becomes more valuable as yields fall.

  Fix: As yields fall, the callable bond's price is capped near the call price, so its effective duration shrinks. It is not above the option-free bond's duration.

• Believing duration falls smoothly to zero as a coupon bond ages.

  Using the zero-coupon pattern for all bonds.

  Fix: At constant yield, duration declines between coupon dates and jumps up right after a coupon payment, with a general downward trend toward zero at maturity.

Worked examples