Common mistake 1 with Bond price, duration & convexity

Quoting duration for a large yield move and ignoring convexity - the linear estimate always overstates the loss on a rise.

Common mistake 2 with Bond price, duration & convexity

Comparing the durations of two bonds with different yields or payment frequencies without noting it; the measure is only locally valid.

Common mistake 3 with Bond price, duration & convexity

Treating duration as protection against any rate move. It only immunises against a parallel shift.

Common mistake 4 with Bond price, duration & convexity

Using modified duration on a bond with an embedded option. A callable bond's price does not move smoothly with yield, and effective duration - computed by repricing - is the right measure.

EconStatsCraft › Bond price, duration & convexity

Bond price, duration & convexity

Prices a coupon bond off its yield (or solves the yield from a price), then reports duration, convexity and DV01 and shows how far the linear duration estimate misses on a 100 bp move.

The intuition

A bond is a fixed stream of future cash. Its price is what that stream is worth today, so when the discount rate rises the price falls - always, and by an amount that depends on how far away the cash is. Duration measures that sensitivity as a single number. Convexity measures how wrong duration is, because the price-yield relationship is a curve and duration is only its tangent.

When to use it

Valuing a fixed-coupon bond, measuring and hedging interest-rate risk, immunising a portfolio against a parallel yield shift, or converting between a bond's price and its yield to maturity.

Where the estimator comes from

Price is P=\sum_t C_t/(1+y)^t. Differentiating, dP/dy=-\sum_t t\,C_t/(1+y)^{t+1}, so the proportional sensitivity is -\frac{1}{P}\frac{dP}{dy}=\frac{1}{1+y}\sum_t t\,w_t with w_t=\frac{C_t/(1+y)^t}{P}. The sum is MACAULAY duration, a weighted average time to payment measured in years; dividing by (1+y) gives MODIFIED duration, which is the percentage price change per unit of yield. The second derivative gives convexity, \mathcal{C}=\frac{1}{P(1+y)^2}\sum_t t(t+1)\,C_t/(1+y)^t, and the second-order approximation is \Delta P/P\approx-D_{mod}\Delta y+\tfrac12\mathcal{C}(\Delta y)^2.

The model

P = \sum_{t=1}^{n}\frac{c}{(1+y)^{t}} + \frac{F}{(1+y)^{n}},\qquad D_{\text{mod}} = \frac{-1}{P}\frac{\partial P}{\partial y},\qquad \Delta P \approx -D_{\text{mod}}\,\Delta y\,P + \tfrac12\,C\,(\Delta y)^2 P

What it assumes

A single flat yield discounts every cash flow - one number, not a term structure.
Cash flows are fixed and certain: no default, no embedded call or put, no floating coupon.
Duration and convexity describe a PARALLEL shift in the yield; a change in the shape of the curve is not captured.
Coupons are reinvested at the yield to maturity. That is what makes the yield the realised return if held to maturity, and it is almost never true - reinvestment risk is the gap between the quoted yield and what you actually earn.

Statistical properties

Duration is a weighted average time, so it is measured in years and is always less than maturity for a coupon bond, and exactly equal to maturity for a zero.
Convexity is always positive for an option-free bond. Prices rise more when yields fall than they fall when yields rise by the same amount, which is a genuine advantage to the holder.
A callable bond can have negative convexity: when yields fall the issuer calls, so the price stops rising. Mortgage-backed securities behave the same way through prepayment.
Duration falls as the coupon rises, because more of the cash arrives sooner. A zero-coupon bond has the highest duration for its maturity.
Duration also measures the horizon at which price risk and reinvestment risk offset. A portfolio with duration equal to the investment horizon is immunised against a small parallel shift.
All of this assumes the whole yield curve moves in parallel. It rarely does, which is why key-rate durations exist.

How to read the output

Price above face value means the coupon exceeds the yield (a premium bond); below means the opposite (a discount bond); equal means it trades at par.
Modified duration ≈ the percentage price fall for a one-percentage-point rise in yield. A duration of 7 means roughly a 7% loss on a +100 bp move - before the convexity correction.
Convexity is always positive for an option-free bond and always helps the holder: it softens losses when yields rise and amplifies gains when they fall.
DV01 (or PVBP) is the cash price change for a one-basis-point move - the unit traders hedge in.
Longer maturity, lower coupon and lower yield all raise duration.
Macaulay duration is in years and is a weighted average time to repayment; modified duration is a percentage price sensitivity. They are close in value and different in meaning, and only one of them belongs in a sentence about price moves.
DV01 is the number a trading desk uses, because it is in cash rather than percent and therefore adds across a portfolio of different-sized positions.

A worked reading

A 10-year bond paying 4% in semi-annual coupons, priced to yield 5%, costs 92.2054 - below par, because the coupon is below the yield. Macaulay duration is 8.2556 years, modified duration 8.0542, convexity 77.1117, and DV01 is 0.0743 per basis point. Now raise the yield by a full percentage point. Duration alone predicts 84.7790; the exact reprice is 85.1225, so the linear estimate understates the bond by 0.3435 - a third of a point, on a position of any size a real number. Adding the convexity term gives 85.1345, wrong by 0.0120, about a penny. That gap is the whole reason convexity is quoted beside duration: at 100 basis points the linear approximation is already visibly off, and it deteriorates with the square of the move, so at 300 basis points it is useless.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Duration longer than maturity	An arithmetic error, unless the bond is leveraged	Recheck the cash flow schedule
Price moves badly mispredicted by duration	The yield change was large	Add the convexity term; duration is a tangent
Negative convexity reported	A call or prepayment feature	Correct behaviour, but the linear approximation is now unsafe in both directions
Duration of a floating-rate note near zero	Correct. It reprices at each reset	Use the time to next reset, not maturity
Yield entered as 6 rather than 0.06	Percent versus decimal	Price will come out absurdly small
Portfolio duration far from the sum of parts	Weights not by market value	Duration aggregates by value weight, not by count

Common mistakes

Quoting duration for a large yield move and ignoring convexity - the linear estimate always overstates the loss on a rise.
Comparing the durations of two bonds with different yields or payment frequencies without noting it; the measure is only locally valid.
Treating duration as protection against any rate move. It only immunises against a parallel shift.
Using modified duration on a bond with an embedded option. A callable bond's price does not move smoothly with yield, and effective duration - computed by repricing - is the right measure.

Where to read this properly

Macaulay (1938), Some Theoretical Problems Suggested by the Movements of Interest Rates - the original.
Fabozzi, Bond Markets, Analysis and Strategies (2021), ch. 4 and 5.
Tuckman and Serrat, Fixed Income Securities (2022), ch. 4, on key-rate durations.

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Bond price, duration & convexity

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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