Duration Hedging a Treasury Book: The Convexity the CIO Forgot
A fixed-income manager oversees a $100 million Treasury portfolio with modified duration 7.2 and convexity 85. She expects rate cuts, but the CIO has ordered the portfolio's duration hedged to zero for two months ahead of an accounting review. She will use 10-year Treasury note futures. The cheapest-to-deliver bond has a DV01 of $72.50 per $100,000 face value and a conversion factor of 0.8125; the futures price is 118-00 and the contract is $100,000 face. The CIO states that "a duration hedge means the book cannot lose money if rates fall or rise."
1. Estimate the portfolio's price change for a 100 bp fall in yields using duration only, then duration plus convexity. Explain the gap.
Read the worked answer
Duration only: ΔP/P ≈ −D_mod × Δy = −7.2 × (−0.0100) = +7.20% → +$7.20 million
Duration plus convexity: ΔP/P ≈ −D_mod × Δy + ½ × C × (Δy)² = 7.20% + ½ × 85 × (0.01)² = 7.20% + ½ × 85 × 0.0001 = 7.20% + 0.425% → +7.625%, or +$7.625 million
The gap is $425,000 - the convexity term. Duration is the first-order (tangent) approximation and is symmetric: it predicts the same magnitude of gain for a 100 bp fall as loss for a 100 bp rise. Convexity is the second-order curvature and is always positive for a non-callable bond: it adds to the gain when yields fall and cushions the loss when yields rise.
For a 200 bp move the approximation error grows roughly fourfold (convexity scales with Δy²): duration predicts ±14.4%, while duration-plus-convexity gives +16.1% on a 200 bp fall and −12.7% on a 200 bp rise. The asymmetry - 16.1% up versus 12.7% down - is convexity, and it is the reason a duration-neutral hedge cannot be a P&L-neutral hedge.
2. How many futures contracts are required to bring portfolio duration to zero? Explain each element of the ratio, including the conversion factor.
Read the worked answer
Step 1 - portfolio DV01 (change in value per 1 bp): DV01_P = D_mod × Value × 0.0001 = 7.2 × $100,000,000 × 0.0001 = $72,000 per bp
Step 2 - futures DV01. The futures tracks the cheapest-to-deliver, but in conversion-factor-adjusted units. The CTD's DV01 is $72.50 per contract face; the futures price behaves like CTD price ÷ conversion factor, so:
DV01_futures = DV01_CTD / CF = 72.50 / 0.8125 = $89.23 per bp per contract
Intuition: an 0.8125 conversion factor means the deliverable is cheaper than the notional bond, so one futures contract gives more DV01 exposure than the CTD's raw DV01 - hedging requires fewer contracts than a naive CTD-only calculation would suggest.
Step 3 - hedge ratio: N = DV01_P / DV01_futures = 72,000 / 89.23 = 806.9 → short 807 contracts
Check: 807 × $89.23 ≈ $72,008 per bp, against a portfolio DV01 of $72,000 - a residual of roughly $8 per bp, which is immaterial.
Why short: the portfolio gains when yields fall; the futures must lose when yields fall, so the position is short. Note the futures price level (118-00) does not enter the ratio directly - only the DV01 does - and the notional being hedged (807 × $100,000 ≈ $80.7 million face, or $95.3 million at the futures price) is not the same as the portfolio's market value. That mismatch is normal and is not an error.
3. Evaluate the CIO's claim using a 200 bp scenario in each direction, and list what the hedge leaves unhedged.
Read the worked answer
200 bp fall (rates down):
- Portfolio: +16.1% ≈ +$16.1 million
- Short futures: 807 × $89.23 × 200 = −$14.40 million
- Net ≈ +$1.7 million
200 bp rise (rates up):
- Portfolio: −12.7% ≈ −$12.7 million
- Short futures: +$14.40 million
- Net ≈ +$1.7 million
So the CIO's claim is wrong in both its parts: the book is not protected against a fall (the hedge loses money there - it is the upside the hedge sells), and the residual is a small convexity gain in both directions, not zero. The correct statement is: "a duration hedge neutralises first-order exposure; the residual is convexity plus basis and operational risk."
What remains unhedged:
- Convexity - small here, large for long-dated or callable paper; it also means the hedge ratio is only valid locally and must be rebalanced as duration drifts.
- CTD switch - if yields move enough, the cheapest-to-deliver bond can change, changing the conversion factor and DV01 and therefore the correct contract count. The 807 is a point-in-time number.
- Basis and roll - futures converge to spot; a 2-month horizon may straddle a roll, and the futures-spot spread can move independently of the portfolio's own spreads.
- Curve risk - the portfolio's 7.2 duration sits across the curve; hedging with the 10-year futures neutralises a parallel shift, not a steepening or flattening. A 2s10s move can leave the book exposed.
- Margin and liquidity - 807 short contracts require variation margin; a rally forces cash outflows at the worst time.
- Accounting asymmetry - the CIO's review is accounting-driven; the hedge may be marked at fair value while the portfolio is held at amortised cost, producing P&L volatility even where economic exposure is neutral. That, not the maths, is often the real constraint, and it should be raised explicitly before the hedge is placed.