16.4 Duration, Convexity & DV01

Duration, Convexity & DV01

Once bonds are priced off a curve, risk asks: how much does value move when rates move? VRM–12 answers with one-factor sensitivities—DV01, duration, and convexity—then applies them to hedge sizing, portfolios, and the classic bullet versus barbell comparison.

One-Factor Models of Rate Risk

A one-factor description assumes a single risk driver—commonly a parallel shift of the yield curve (every yield ±Δy) or a shift in a short rate that correlates all tenors. Under that assumption, bond price P(y) is locally a function of one variable y, and Taylor expansion gives

ΔP ≈ (dP/dy) Δy + (1/2) (d²P/dy²) (Δy)² + …

First derivative → duration / DV01. Second → convexity. Real curves also twist and butterfly; one-factor hedges neutralize only the chosen factor. Still, GARP Part I leans heavily on parallel DV01 intuition because it is the industry’s first line of defense.

DV01 Calculation and Hedge Sizing

DV01 (dollar value of 01) is the change in position value for a 1 basis point increase in yield (sign conventions vary; many desks quote positive DV01 as the absolute money risk per bp).

Analytic style for a long bond:

DV01 ≈ −(dP/dy) × 0.0001

so if modified duration D_mod = 7.0 and dirty value V = $10,000,000,

ΔV ≈ −D_mod × V × Δy

For Δy = +0.0001 (1 bp), ΔV ≈ −7 × 10m × 0.0001 = −$7,000. DV01 ≈ $7,000 per bp (absolute).

Hedge sizing against hedge instrument H:

N_H ≈ − DV01_portfolio / DV01_per_unit_H

If a CTD futures (or a 10y note) has DV01 = $90 per contract per bp and the book’s DV01 is +$450,000 per bp (long rates risk), short N ≈ 450,000/90 = 5,000 contracts to flatten one-factor exposure.

Worked hedge

Asset DV01 = +$25,000 / bp. Hedge bond DV01 = +$125 / bp per $100k face. Face to short = 25,000/125 × 100k = $20,000,000 face. After the hedge, first-order parallel risk ≈ 0; convexity and spread risks remain.

Effective Duration

Macaulay duration is the PV-weighted average maturity of cash flows. Modified duration = Macaulay / (1 + y/m) and scales percentage price change versus yield. Both assume fixed cash flows.

Effective duration estimates sensitivity when cash flows may change with rates (callables, MBS):

D_eff = [P(−Δy) − P(+Δy)] / [2 × P0 × Δy]

using full pricing models at shifted curves. For option-free bonds, effective ≈ modified. For callables near the call region, effective duration collapses relative to maturity because price is capped by the call.

Worked effective duration

P0 = 98.00, P(down 10 bp) = 98.70, P(up 10 bp) = 97.35, Δy = 0.001.

D_eff = (98.70 − 97.35) / (2 × 98.00 × 0.001) = 1.35 / 0.196 = 6.89.

DV01 Versus Duration

Duration is a percentage (or time) sensitivity; DV01 is a money sensitivity.

ΔP/P ≈ −D_mod × Δy

DV01 ≈ D_mod × P × 0.0001 (absolute dollars per bp for long positions)

Same information for a single position if you know P; DV01 aggregates additively across positions in dollars, which is why books and hedges speak DV01. Duration aggregates only with value weights, not face weights.

Convexity

Convexity captures curvature:

Conv ≈ [P(−Δy) + P(+Δy) − 2 P0] / [P0 × (Δy)²]

Price approximation:

ΔP/P ≈ −D_mod Δy + (1/2) Conv (Δy)²

For option-free bonds, convexity is positive: gains from a large yield drop exceed losses from an equal yield rise (price is convex in yield). That asymmetry is valuable when volatility is high. Negative convexity appears in callables/MBS when rates fall and prepayment/call risk shortens the instrument.

Worked convexity add-on

D_mod = 6.89, Conv = 55, Δy = −0.01 (−100 bp). Duration term = −6.89×(−0.01) = +6.89%. Convexity term = 0.5×55×(0.01)² = +0.275%. Approximate return ≈ +7.165% versus +6.89% from duration alone.

Portfolio Measures

For positions i with values V_i, modified durations D_i, convexities C_i:

D_port = Σ (V_i / V) D_i

C_port = Σ (V_i / V) C_i

DV01_port = Σ DV01_i

Always use dirty market values for weights. Mixing a 2y and a 30y to match a 10y duration creates a barbell with higher convexity than a bullet 10y—same duration, different second-order and twist risk.

Duration + Convexity Hedging

A single hedge instrument can match DV01 but not convexity. Using two hedges (e.g., 2y and 30y vs a 10y liability) you solve two equations:

N1 × DV01_1 + N2 × DV01_2 = −DV01_target

N1 × Conv$_1 + N2 × Conv$_2 = −Conv$_target

(with convexity in dollar terms). The solution often resembles a barbell hedge of a bullet liability—or vice versa.

Limits: still one-factor (or two moment) thinking; key-rate exposures and spread basis survive.

Bullet Versus Barbell

• Bullet: cash flows concentrated near one maturity (e.g., a 10y bond). Lower convexity for a given duration than a barbell.
• Barbell: combination of shorter and longer bonds with the same duration as the bullet. Higher convexity → better performance for large parallel moves if both are option-free and priced fairly on a flat-curve one-factor world.

If the curve steepens or flattens, the barbell’s long and short wings move differently; the bullet can outperform despite lower convexity. Convexity is not free: barbells often cheap or rich on the curve relative to bullets (butterfly richness).

Worked qualitative comparison

Liability duration = 8. Bullet asset: 8y note. Barbell: 2y + 30y weights chosen so portfolio duration = 8 and value matches. Large parallel rally: barbell’s extra convexity tends to win. Pure steepener with front-end rallying and long end selling off: barbell can lose to the bullet.

Exam checklist

1. State the one-factor shock (parallel bp).
2. Compute DV01 or effective duration from analytic or reval formulas.
3. Size hedges with N = −DV01_p / DV01_h.
4. Add convexity for large moves and for bullet/barbell questions.
5. Remember duration match ≠ curve-shape match.

Master DV01 arithmetic and the Taylor story, and you can attack the bulk of VRM rate-risk items without a full multi-factor engine.