Macaulay vs Modified vs Effective Duration: Which One the Question Wants

Duration is the most heavily examined concept in Level 1 Fixed Income, and most lost marks come not from arithmetic but from selection. A candidate who computes modified duration flawlessly on a callable bond has produced a correct number and a wrong answer.

The three measures

Macaulay duration

The weighted average time until a bondholder receives the bond's cash flows, with each cash flow's present value as its weight. It is measured in years.

For a zero-coupon bond, Macaulay duration equals time to maturity exactly — there is only one cash flow, so the weighted average is that single date. This is a favourite exam shortcut: a five-year zero has a Macaulay duration of 5.

For a coupon bond, Macaulay duration is always less than maturity, because some cash flow arrives earlier than the final payment. The higher the coupon, the earlier the weighted average arrives, and the lower the duration.

Modified duration

ModDur = Macaulay duration / (1 + periodic yield)

This converts a time measure into a sensitivity measure. Modified duration estimates the percentage change in a bond's price for a 1% (100 basis point) change in its yield.

A modified duration of 6.2 means a 100 basis point rise in yield produces an approximate 6.2% fall in price. Note that modified duration is always slightly smaller than Macaulay duration, because you are dividing by a number greater than 1.

Effective duration

EffDur = (PV₋ − PV₊) / (2 × Δcurve × PV₀)

Where PV₋ is the price if the benchmark curve falls, PV₊ the price if it rises, and PV₀ the current price.

The critical difference: effective duration measures sensitivity to a shift in the benchmark yield curve, not to the bond's own yield to maturity, and it is computed by repricing the bond under each scenario rather than from a formula about timing.

The rule that decides which to use

If the bond has an embedded option, you must use effective duration.

The reason is that Macaulay and modified duration both assume the cash flows are fixed and known. For a callable bond, they are not. If rates fall far enough, the issuer calls the bond and the remaining cash flows disappear. For a putable bond, the holder may put it back. The cash flow schedule is itself a function of the interest rate path, so any measure derived from a fixed schedule is invalid.

Any bond whose cash flows can change with rates — callable, putable, mortgage-backed securities with prepayment risk, floating-rate notes — requires effective duration.

Recognition cue: the words "callable", "putable", "prepayment", or "embedded option" anywhere in the question mean effective duration, regardless of what data you were given.

The behaviour of callable bonds

This is where the concept becomes genuinely interesting, and where the exam tests understanding rather than recall.

As yields fall, an option-free bond's price rises without limit. A callable bond's price rise is capped, because the market knows the issuer will call it at the call price. The price-yield relationship flattens — the bond exhibits negative convexity in that region.

Two consequences the exam tests:

• A callable bond's effective duration falls as rates fall, because the call becomes more likely and the effective maturity shortens.
• Negative convexity means the price gain from a rate fall is smaller than the price loss from an equivalent rate rise — the reverse of the favourable asymmetry an option-free bond enjoys.

A putable bond behaves in the opposite direction: the put sets a price floor, and effective duration falls as rates rise.

Putting duration to work

The formula that appears in some form on nearly every Level 1 exam:

%ΔPrice ≈ (−ModDur × Δy) + (0.5 × Convexity × Δy²)

Worked example. A bond has a modified duration of 7.5 and convexity of 68. Yields rise by 50 basis points.

Duration effect: −7.5 × 0.005 = −3.75%
Convexity effect: 0.5 × 68 × 0.005² = 0.5 × 68 × 0.000025 = +0.085%
Total: −3.665%

Now the same bond with yields falling 50 basis points:

Duration effect: −7.5 × (−0.005) = +3.75%
Convexity effect: 0.5 × 68 × (−0.005)² = +0.085%
Total: +3.835%

The trap: the convexity term is positive in both cases, because Δy is squared. For an option-free bond convexity always helps — it reduces the loss when rates rise and increases the gain when they fall. Candidates who apply the negative sign to both terms get the falling-rate case wrong, and the wrong answer will be among the choices.

Related measures worth knowing

• Money duration = annual modified duration × full price of the position. Expresses sensitivity in currency rather than percentage terms.
• Price value of a basis point (PVBP) = money duration × 0.0001. The currency price change for a one basis point move.
• Key rate duration measures sensitivity to a change at one point on the curve while holding others constant — useful when the curve twists rather than shifts in parallel.
• Spread duration measures sensitivity to a change in credit spread rather than in the benchmark rate.

What drives duration

Worth knowing directionally, because questions often ask you to rank bonds without calculating:

• Longer maturity → higher duration. Cash flows arrive later.
• Higher coupon → lower duration. More value arrives earlier.
• Higher yield → lower duration. Distant cash flows are discounted more heavily, shifting the weighted average earlier.

So the highest-duration bond in a set is typically the longest-dated, lowest-coupon, lowest-yielding one — and a long-dated zero-coupon bond is the extreme case. Full treatment of the topic sits in the Fixed Income guide, and the formulas are listed in the formula sheet.

Exam checklist

• Embedded option anywhere in the question → effective duration, no exceptions
• Macaulay is in years; modified is a percentage sensitivity
• Modified duration is always slightly below Macaulay
• Zero-coupon bond: Macaulay duration equals maturity
• The convexity term is positive whichever way yields move
• Callable bonds show negative convexity when rates fall