Tailing the Hedge

Q: Tailing the hedge typically results in:

Fewer futures contracts than the naive count. The tail factor e^-rT is less than one for r > 0 and T > 0, so the corrected count N^*_tailed < N_naive. Daily mark-to-market gains can be reinvested at the risk-free rate, which means a naive hedger using Q_A/Q_F would systematically over-hedge unless this correction is applied.

Q: Which of the following best explains why tailing is unnecessary for a forward contract?

Forwards have no daily mark-to-market settlement. Forwards exchange a single cash flow at maturity, so there is no interim variation margin to reinvest. The reinvestment bias that motivates tailing only exists for futures and other daily-settled contracts. Option A is false in general, option C is wrong because forwards still have basis risk if the underlying differs, and option D is irrelevant to the tail.

Tailing the hedge adjusts the futures position downward to account for the time value of daily mark-to-market cash flows. A futures hedge pays or receives variation margin every day, so gains earn interest before they are needed and losses must be funded earlier than under a forward. The naive minimum-variance contract count slightly over-hedges in dollar terms. Multiplying by a tail factor of $e^{-rT}$ , or equivalently scaling by the spot-to-futures ratio $V_A / V_F$ , removes the bias.

Try it yourself

Minimum-variance hedge ratio

You hold an asset and short futures to offset its price moves. The hedge ratio h is how many units of futures you short per unit of exposure. The variance of the hedged position is a parabola in h, minimised at h* = ρ · σ_spot / σ_futures. The deeper the correlation, the more variance the hedge removes — exactly ρ² of it.

Hedge effectiveness 64%. At h* the hedge removes 64% of the unhedged variance, leaving 0.52 of residual (basis) risk you cannot diversify away.

Hedge ratio h*0.96

Optimal contracts N* (exact)19.2

N* rounded to whole contracts19

Hedge effectiveness ρ²64%

Unhedged variance σ_spot²1.44

Minimum variance at h*0.52

Correlation ρ0.8

σ_spot (spot change std dev)1.2

σ_futures (futures change std dev)1

Exposure Q_A (units to hedge)10,000

Contract size Q_F (units per contract)500

h* = ρ · σ_spot / σ_futures · N* = h* · Q_A / Q_F · Var(h) = σ_spot² + h²·σ_futures² − 2·h·ρ·σ_spot·σ_futuresσ_spot and σ_futures are standard deviations of price changes over the hedge horizon. Effectiveness ρ² is the share of spot variance the hedge removes; the rest is basis risk. Contracts are discrete, so N* is rounded to whole contracts in practice.

Why it matters

Forwards settle one cash flow at maturity. Futures pay you (or charge you) daily. A futures gain received six months early can be invested and grown at the risk-free rate, so it is worth more than the same gain at maturity. The hedger therefore needs slightly fewer futures than a forward hedge would suggest. The correction is small but matters for institutional hedges running into the hundreds of millions of dollars.

Before you read on — recall

Tailing the hedge typically results in:

Formulas

Naive hedge contract count

N_{\text{naive}} = h^* \times \frac{Q_A}{Q_F}

Treats futures as if they settled only at maturity, ignoring daily mark-to-market.

Tailed contract count (Hull form)

N^* = h^* \times \frac{V_A}{V_F}

V_A = S_t \times Q_A

is the dollar value of the spot exposure and

V_F = F_t \times Q_F

is the dollar value of one futures contract. The ratio

V_A / V_F

replaces

Q_A / Q_F

and embeds the spot-futures price adjustment. Source: Hull (2022) §3.5.

Continuous-compounding equivalent

N^*_{\text{tailed}} \approx N_{\text{naive}} \times e^{-rT}

Holds approximately when

S \approx F

and

r

is the risk-free rate over the hedge horizon

T

. The two forms agree to first order.

Worked examples

Scenario

A copper smelter has a long hedge with $h^* = 0.95$ . Spot copper is US$8,000 per tonne, futures are US$8,080 per tonne, the position size is $Q_A = 12{,}500$ tonnes, and each LME contract covers $Q_F = 25$ tonnes.

Solution

$V_A = 12{,}500 \times 8{,}000 =$ US$100,000,000. $V_F = 25 \times 8{,}080 =$ US$202,000. Tailed contracts $N^* = 0.95 \times (100{,}000{,}000 / 202{,}000) = 470.3$ , rounded to 470. The naive count 0.95 × (12,500 / 25) = 475 would over-hedge by 5 contracts. Here the tail of about 1 percent comes from the spot-to-futures ratio $V_A/V_F = S/F = 8{,}000/8{,}080 = 0.990$ , the $V_A/V_F$ scaling, which is the carry over the futures' life and is not the same quantity as the $e^{-rT}$ factor over the hedge horizon (the next example shows the $e^{-rT}$ form).

Scenario

Naive contract count is 100, risk-free rate $r = 0.05$ , hedge horizon $T = 0.5$ years.

Solution

Tail factor $e^{-rT} = e^{-0.025} = 0.9753$ . Tailed contracts $N^*_{\text{tailed}} = 100 \times 0.9753 = 97.53$ , rounded to 98. The 2 to 3 contract reduction prevents the reinvestment of mark-to-market gains from causing systematic over-hedging.

Common mistakes

✗Tailing produces large changes in contract numbers. The adjustment is typically 1 to 5 percent, depending on rates and horizon. It matters most for large institutional hedges and longer maturities, where small percentages translate into millions of dollars.
✗Tailing applies to forwards as well. Forwards settle once at maturity, so there is no daily mark-to-market to reinvest. The tail adjustment is specific to futures and other daily-settled contracts.
✗The tail factor uses the maturity of the futures contract. The relevant horizon is the hedge horizon $T$ , that is, how long the hedger expects to hold the position, not the futures expiry. For overnight hedges, the tail effect is negligible.

Revision bullets

•Adjusts for daily mark-to-market cash flows
• $N^* = h^* \times V_A / V_F$ in Hull's form
•Equivalent to multiplying by $e^{-rT}$
•Reduces the naive futures position
•Larger effect when rates or horizons are big
•Does not apply to forwards

Quick check

Tailing the hedge typically results in:

Which of the following best explains why tailing is unnecessary for a forward contract?

Connected topics

Mark-to-Mkt Basis Risk Min Var. HR

More in Hedging & Basis Risk

Hedge vs Spec Short Hedge Long Hedge Beta Hedge Change Beta Basis Cross Hedge

In learning paths

Hedging & Risk Path

Sources

Hull (2022), §3.5
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Section 3.5 introduces the tailing adjustment, derives the $V_A / V_F$ form, and contrasts the futures hedge with the equivalent forward.
Figlewski, Landskroner & Silber (1991)
Figlewski, Stephen, Yoram Landskroner, and William L. Silber. Tailing the Hedge: Why and How. Journal of Futures Markets, Vol. 11, No. 2, 1991, pp. 201-212.
Original article that formalised the tailing adjustment and quantified its impact across different hedge horizons and interest-rate environments.
CME Group, Treasury Hedging
CME Group. Treasuries Hedging and Risk Management. CME Education Centre, accessed 2026.
Practitioner reference describing how dealers adjust hedge ratios for daily settlement in interest-rate futures, where tailing is most material.

How to cite this page

Dr. Phil's Quant Lab. (2026). Tailing the Hedge. Derivatives Atlas. https://phucnguyenvan.com/concept/tailing-the-hedge