Finance Formulas

Fixed Income

core

Modified Duration

Builds onMacaulay Duration — if this page feels steep, start there.

ModDur=MacDur1+yModDur = \frac{MacDur}{1+y}

Reading the notation

ModDurModDur
modified duration: % price change per 1-percentage-point yield change
MacDurMacDur
Macaulay duration: the value-weighted average time to the cash flows, in years
1+y1+y
one period's growth factor at the yield — the discrete-compounding correction

Play with it

The dashed tangent line IS modified duration: the straight-line price response at the current yield. Move the yield marker and the tangent re-anchors.

$0$500$1000$15002%5%8%11%14%yieldpriceactual priceduration tangent

$926

Price

8.02y

Macaulay dur.

7.57

Modified dur.

73

Convexity

ScenarioDuration-only est.Actual priceConvexity gap
-200bp$1067$1081+$14
-100bp$997$1000+$3
+100bp$856$860+$3
+200bp$786$799+$13

The gap is positive in BOTH directions — the curve always bows above its tangent. That bow is convexity.

Why it must be true

Macaulay duration is a time (years to get your money back, on average). Divide it once by (1+y)(1+y) and it becomes a sensitivity: the percentage the price moves for a 1-point move in yield. A modified duration of 6.8 means a 1% yield rise knocks roughly 6.8% off the price.

Why does time convert into sensitivity at all? Because discounting punishes distant cash flows hardest — the longer your money is out (bigger MacDur), the more a yield change compounds against it. The (1+y)(1+y) divisor is just the calculus adjustment from discrete compounding.

The derivation

Differentiate the price with respect to yield. Each term CFt(1+y)t\frac{CF_t}{(1+y)^t} has derivative −t⋅CFt(1+y)t+1\frac{-t \cdot CF_t}{(1+y)^{t+1}}, so:

dPdy=−11+y∑t⋅CFt(1+y)t=−MacDur⋅P1+y\frac{dP}{dy} = \frac{-1}{1+y}\sum t \cdot \frac{CF_t}{(1+y)^t} = \frac{-MacDur \cdot P}{1+y}

Divide by −P-P to state it as a percentage sensitivity:

ModDur=−1PdPdy=MacDur1+yModDur = -\frac{1}{P}\frac{dP}{dy} = \frac{MacDur}{1+y}

When to reach for it

Converting a Macaulay duration into price sensitivity, or any 'how much does the price move per unit of yield' question.

Listen for

“modified duration”“price sensitivity to yield”“Macaulay duration of … yield of …”“% change per 1% change in yield”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

  • ≈

    Modified is always a shave BELOW Macaulay — by y percent, roughly. MacDur 7.0 at a 6% yield → ModDur ≈ 6.6. If a choice exceeds MacDur, it's wrong on sight.

  • ≈

    At low yields the two nearly coincide (divide by 1.02 barely moves it) — don't burn time on precision the choices don't require.

Traps in applying it

  • ✗Multiplying by (1+y) instead of dividing — sensitivity is always ≤ the time measure.
  • ✗Semiannual bonds: divide by (1 + y/2), the PERIODIC yield, not the annual one.
  • ✗Quoting Macaulay to a risk system expecting modified — the numbers are close enough to slip through review and wrong enough to mis-hedge.

Limits & criticisms

Modified duration is the slope of the price-yield curve at one point — a straight-line estimate that assumes small, parallel yield moves and fixed cash flows. For big moves the curve bends away from the tangent (convexity), for callable bonds the cash flows themselves shift (use effective duration), and for non-parallel curve twists you need key-rate durations. It's the first derivative of a curved world.

Where it came from

The conversion from Macaulay's 1938 time-measure into a price sensitivity came with John Hicks (1939), who derived the same quantity independently as an elasticity — which is why some texts call it "Hicks duration." The two strands merged in the 1970s rate storms, when Salomon Brothers' research desk (Martin Leibowitz) turned duration into the trading floor's everyday risk language.

Today "duration" on any desk means THIS number: portfolio durations are targeted, hedged and reported daily, and central banks talk about the duration risk sitting in bank balance sheets.

One identity, 2 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

Time into sensitivity

ModDur=MacDur1+yModDur = \frac{MacDur}{1+y}

The everyday direction: rescale the average-time statistic into % price change per point of yield.

Drill this face →

Recover the time measure

MacDur=ModDur×(1+y)MacDur = ModDur \times (1+y)

Backwards: a risk system reports modified duration; immunization needs Macaulay. Multiply the (1+y) back on.

Drill this face →

On the BA II Plus

Worked example: A bond's Macaulay duration is 9.25 years at a yield of 2.5%. What modified duration should the risk report show?

  1. 1.9.25 [÷] 1.0250 [=]Macaulay ÷ (1 + y)

→ 9.0244

Where it leads

Master this and the following come almost for free:

Money Duration & PVBPPrice Change from Duration & Convexity
Drill this formulaBack to the library