Finance Formulas

Fixed Income

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Price Change from Duration & Convexity

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

ΔPP≈−ModDur⋅Δy+12⋅Conv⋅(Δy)2\frac{\Delta P}{P} \approx -ModDur \cdot \Delta y + \tfrac{1}{2} \cdot Conv \cdot (\Delta y)^2

Reading the notation

ΔPP\frac{\Delta P}{P}
the percentage change in the bond's price (Δ means 'change in')
ModDurModDur
modified duration — the straight-line % sensitivity per point of yield
Δy\Delta y
the yield change, as a decimal (a 50bp rise is +0.005)
−ModDur⋅Δy-ModDur \cdot \Delta y
the tangent-line estimate: yields up → price down, scaled by duration
12Conv(Δy)2\tfrac{1}{2}Conv(\Delta y)^2
the curvature correction — squared Δy is always positive, so it always helps (for ordinary bonds)

Play with it

The formula's two terms, drawn: the tangent (duration) plus the bow between tangent and curve (convexity). Check the table — the gap is positive in both directions.

$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

The price-yield relationship is a curve, and duration is only its tangent line. For small yield moves the tangent is fine; for big ones it always errs the same way — because the curve bows away from the line in the bond-holder's favor. Prices fall less than duration predicts when yields rise, and gain more when yields fall.

Convexity is the size of that bow, and the 12(Δy)2\tfrac{1}{2}(\Delta y)^2 term adds it back. Since (Δy)2(\Delta y)^2 is positive whichever way yields move, positive convexity is a free lunch of sorts — which is why investors pay up for it, and why the interactive price-yield model on this page is worth playing with: drag the yield and watch the gap between line and curve open.

The derivation

Taylor-expand the price around the current yield — the same second-order expansion as anywhere in calculus:

ΔP≈dPdyΔy+12d2Pdy2(Δy)2\Delta P \approx \frac{dP}{dy}\Delta y + \tfrac{1}{2}\frac{d^2P}{dy^2}(\Delta y)^2

Divide by PP and name the derivatives: −1PdPdy-\frac{1}{P}\frac{dP}{dy} is modified duration and 1Pd2Pdy2\frac{1}{P}\frac{d^2P}{dy^2} is convexity:

ΔPP≈−ModDur⋅Δy+12 Conv (Δy)2\frac{\Delta P}{P} \approx -ModDur \cdot \Delta y + \tfrac{1}{2}\,Conv\,(\Delta y)^2

First term: the tangent line. Second term: the bow of the curve.

When to reach for it

Estimating a bond's percentage price move for a given yield shift when duration (and possibly convexity) are quoted.

Listen for

“yields rise/fall by … basis points”“duration of … convexity of …”“estimated percentage price change”“second-order / convexity adjustment”

Back-of-the-envelope

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

  • ≈

    Duration term first, in your head: ModDur 7 × 75bp = −5.25%. The convexity add-back is small: ½ × 150 × (0.0075)² ≈ +0.42%. The answer sits just ABOVE the pure-duration number.

  • ≈

    Sign discipline: the duration term flips sign with Δy; the convexity term NEVER does (positive convexity always adds). Two candidate answers symmetric around the duration estimate? Take the higher one.

  • ≈

    Basis points to decimal without error: 125bp = 0.0125. Squaring kills mistakes — (0.0125)² ≈ 0.00016, tiny.

Traps in applying it

  • ✗Leaving Δy in percent instead of decimal — the squared term then explodes by 10,000×.
  • ✗Adding the convexity term with Δy's sign — it takes the SQUARE of Δy, so it's always positive for ordinary bonds.
  • ✗Using Macaulay instead of modified duration in the first term.

Limits & criticisms

Still a local approximation: a two-term Taylor series around today's yield, assuming a parallel shift and unchanged cash flows. Very large moves need full repricing; curve twists need key-rate durations; and callable/MBS bonds can have NEGATIVE convexity, where the "correction" hurts instead of helps — the formula's sign conventions hold, but the free lunch doesn't.

Where it came from

Duration hedging alone repeatedly disappointed traders on big rate moves, and by the early 1980s — as Salomon Brothers, Leibowitz and the immunization literature professionalized bond math during the Volcker rate shocks — the second-order term became standard practice. "Positive convexity" entered the trading vocabulary, and mortgage desks learned its dark twin: negative convexity, where homeowner refinancing makes MBS prices bow AGAINST the holder.

Today duration-plus-convexity is how every risk system translates a rate scenario into P&L, and convexity itself is actively bought and sold.

One identity, 1 questions

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

Rate scenario → P&L

ΔPP≈−ModDur Δy+12Conv Δy2\frac{\Delta P}{P} \approx -ModDur\,\Delta y + \tfrac{1}{2}Conv\,\Delta y^2

The risk-system direction: turn a yield scenario into an estimated percentage price move — tangent line plus bow.

Drill this face →

On the BA II Plus

Worked example: A risk report shows modified duration 8.5 and convexity 50. Estimate the percentage price impact of a 50bp drop in yield, including the convexity adjustment.

  1. 1.8.5 [×] -0.005 [=] [+|-]the duration (tangent-line) term, sign flipped
  2. 2.[STO] 1park it
  3. 3.-0.005 [x²] [×] 50 [×] 0.5 [=]the convexity correction (always positive here)
  4. 4.[+] [RCL] 1 [=]add the two terms (decimal = % move)

→ 4.31%

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