Linear vs Inverse Perpetual Contracts: P&L, Coin Collateral and Hidden Exposure

Editorial responsibility: Oliver Sinz · Last verified: October 14, 2026

A bitcoin perpetual can pay you in stablecoins or in bitcoin. That looks like a settlement detail, but it changes how profit is computed, what your collateral is worth, how funding is charged and whether a "hedged" book is actually flat.

This guide derives both P&L formulas from first principles and works through three verified examples. The core result: a properly sized linear position and an inverse position can produce the same dollar profit on the same price move, yet deliver it in a different currency, against differently behaving collateral, with different portfolio-level risk.

All prices, positions and rates below are hypothetical illustrations, not market data. Venue specifications were checked on October 14, 2026, and can change.

Linear, inverse, coin-margined: three labels, three questions

These labels are often used interchangeably, but they answer different questions:

• Payoff structure. A linear contract fixes a quantity of the base asset and pays the price change in the quote currency, usually USDT or USDC. An inverse contract fixes a dollar face value per contract and pays in the underlying coin.
• Settlement currency. The currency in which realized P&L lands: stablecoin-settled or coin-settled.
• Collateral currency. The asset that backs the position: stablecoin-margined or coin-margined.

The common pairings are not universal. Bybit's inverse contract introduction explains that in its Unified Trading Account cross- and portfolio-margin modes, assets such as USDT can collateralize an inverse position whose P&L still settles in the coin. Binance's USDⓈ-M vs COIN-M overview lists stablecoins "and other available coins" as collateral for its linear contracts. A quanto contract settles in one coin while paying linearly on another asset's price, so it is coin-settled but not inverse; Ackerer, Hugonnier and Jermann treat linear, inverse and quanto perpetuals as distinct pricing problems.

Here, "inverse" always refers to the payoff, and each example states its collateral explicitly.

What "one contract" means

The same dollar exposure requires very different contract counts depending on the instrument:

Sources: Binance (100 USD per BTC COIN-M contract); Deribit (10 USD, noting that an API contract size of 10 means 10 USD, not 10 BTC); Bybit's introduction, where 70,000 BTCUSD contracts at $35,000 equal 2 BTC, implying 1 USD each.

Line-ups also change. BitMEX, a long-standing inverse venue, ceased trading services on September 23, 2026, and as of October 20, 2026, Kraken's inverse contract page carried a notice scheduling the delisting of its inverse futures for October 30, 2026 at 16:00 UTC, with linear contracts unaffected. Check the current specification before relying on any of these numbers.

Deriving the two P&L formulas

Notation: s = +1 for a long and −1 for a short, q = number of contracts, P₀ = entry price, P₁ = exit price.

Linear. With M units of base asset per contract:

PnL_linear = s × q × M × (P₁ − P₀)

Units: contracts × BTC per contract × USDT per BTC = USDT. Equal dollar moves up and down produce equal gains and losses.

Inverse. With face value F per contract and total face N = q × F, read a long as selling N dollars for bitcoin at entry. That sale is worth N / P₀ BTC. Closing means buying back the same N dollars, which costs N / P₁ BTC. The difference is the profit in coin:

PnL_inverse = s × N × (1/P₀ − 1/P₁)

Units: USD × BTC per USD = BTC. This is the formula in Bybit's inverse P&L documentation and Binance's P&L guide, and the design predates today's venues: Bragin (2015) traces inverse futures to the early bitcoin exchange ICBIT. A long gains when the price rises because 1/P₁ falls below 1/P₀; a short reverses the sign.

The bridge. Multiply the inverse result by the exit price:

PnL_inverse × P₁ = s × (N / P₀) × (P₁ − P₀)

This is the linear formula with q × M = N / P₀. An inverse position valued at the price at which its coin P&L is realized produces the same dollar P&L as a linear position of N / P₀ BTC. It is an identity, not an approximation, provided one consistent price is used for the contract and the valuation.

In practice that price differs by context. Binance computes unrealized P&L on the mark price and realized P&L on the last price, while Bybit's interface shows unrealized inverse P&L on the last traded price by default. The examples below assume a single price for marking, execution and spot valuation.

Example 1: matched linear and inverse positions

Assumptions: Entry $60,000. The linear position is long 0.5 BTC on a USDT-settled perpetual with USDT collateral, valued at $1 per USDT. The inverse position is long $30,000 of face value (300 contracts of $100) with BTC settlement and BTC collateral. Both have an initial notional of $30,000. Exits at $66,000 (+10%) and $54,000 (−10%). No fees, funding or slippage.

The dollar outcomes are identical. The coin cash flows are not: the 10% decline costs 0.0556 BTC while the 10% rise earns 0.0455 BTC, a loss about 22% larger in coin terms. That asymmetry is not a hidden dollar cost, because each coin amount is worth exactly $3,000 at the price at which it is paid. Shorts mirror the result: +$3,000 at $54,000 and −$3,000 at $66,000 under both structures.

The valuation trap. Translating the coin gain at the entry price gives 0.0454545 × $60,000 = $2,727, understating the profit by $273. That gap is not trading P&L; it is bitcoin revaluation between two dates, mixed into the result. Coin received at exit is valued at the exit price, and what happens to it afterward is a separate exposure. Likewise, an inverse profit does not "pay twice" in a rally: the trade earns what a matched linear trade earns, and any further gain comes from continuing to hold the coin.

Why the coin P&L curves

Holding the same $30,000 inverse long from $60,000 across a wider price grid shows the shape:

Measured in bitcoin, a long's P&L is concave in price (its second derivative is −2N/P³), and a short's is convex. Measured in dollars at the same price, it is a straight line, matching a linear position of 0.5 BTC.

The last column explains why. The position value in BTC, N / P, shrinks as the price rises, but the position's coin P&L accumulates in bitcoin:

N/P + N × (1/P₀ − 1/P) = N/P₀

Counting its own unrealized coin P&L, the position's total bitcoin exposure stays at 0.5 BTC. Reading N / P alone as "the exposure" misstates the position whenever unrealized P&L exists.

Example 2: coin collateral is a second position

Assumptions: The same $30,000 inverse long at $60,000 sits in an isolated account funded with 0.25 BTC ($15,000), so face value is twice the collateral. Exits at $72,000 and $48,000. No fees or funding. Account value in USD equals the BTC balance times the price.

Total performance reconciles in two parts:

ΔW = C × (P₁ − P₀) + PnL_inverse × P₁

The first term is collateral revaluation; the second is the derivative's P&L at the exit price. Each is counted once. Converting the final balance to dollars already includes both, so adding the derivative P&L on top would double-count it.

A trader with the same $15,000 in USDT and a 0.5 BTC linear long would gain or lose $6,000, or 40%. The extra $3,000 comes entirely from the 0.25 BTC of collateral. Total dollar exposure is 0.5 + 0.25 = 0.75 BTC, so for someone who measures wealth in dollars, a position sized at two times collateral behaves like three times.

The downside path changes too. In a simplified model with no maintenance margin, fees or funding, the inverse account's equity reaches zero at N / (C + N/P₀) = 30,000 / 0.75 = $40,000, while the USDT-collateralized linear account reaches zero at $30,000. Real liquidation happens earlier because of maintenance requirements, and exact thresholds are venue-specific; a separate guide covers liquidation mechanics.

Funding: same rate, different currency and base

Both structures apply the funding rate to position value, but they measure it differently. Per Bybit's funding fee calculation, linear position value is quantity × mark price, paid in USDT or USDC, while inverse position value is contracts ÷ mark price, paid in the coin. When the rate is positive, longs pay shorts.

With a hypothetical rate of +0.01% per interval:

• Mark $60,000: the $30,000 inverse long pays 0.5 BTC × 0.01% = 0.00005 BTC ($3.00); the 0.5 BTC linear long pays 3.00 USDT.
• Mark $48,000: the inverse long pays 0.625 BTC × 0.01% = 0.0000625 BTC ($3.00); the linear long pays 2.40 USDT.

Inverse funding is constant in dollars per unit of rate and variable in coin; linear funding is the reverse. Funding received on an inverse short arrives in bitcoin and adds bitcoin exposure. Funding paid shrinks the coin balance, and both Bybit and Binance state that funding can be taken from position margin when the available balance is insufficient, which moves liquidation closer. Funding rates vary and cannot be treated as predictable income; a separate guide covers how they are set.

Example 3: the hedge that isn't flat

Assumptions: An investor holds 1.0 BTC in a spot wallet that cannot serve as margin. They deposit a separate 0.2 BTC into an isolated inverse account and short $60,000 of face value at $60,000, which equals 1.0 BTC at entry. The book looks matched: +1 BTC spot against a −1 BTC-equivalent short. Initial wealth is 1.2 BTC × $60,000 = $72,000. Spot and mark prices are equal; no fees, funding or basis. A short's coin P&L is N × (1/P₁ − 1/P₀).

The portfolio still moves $0.20 for every $1 move in bitcoin. Writing wealth as a function of price makes the source explicit:

W(P) = S × P + [C + N × (1/P − 1/P₀)] × P = (S + C − N/P₀) × P + N

dW/dP = S + C − N/P₀ = 1 + 0.2 − 1 = 0.2 BTC

Check: 0.2 × ±$12,000 = ±$2,400, matching the table. Under these static assumptions, the relationship holds exactly for as long as the derivatives account remains solvent, here below $75,000. The residual exposure is the collateral itself: matching the spot holding against the short's BTC-equivalent ignored that the margin is also bitcoin.

Sizing the short at P₀ × (S + C) = $72,000 sets dW/dP to zero. At $48,000, the account holds 0.5 BTC ($24,000) beside $48,000 of spot; at $66,000, it holds 0.0909 BTC ($6,000) beside $66,000 of spot. Either way, total wealth stays at $72,000. This is the coin-collateralized "synthetic dollar" position.

USD neutrality is not the same as safety, though. In the original $60,000 short, the derivatives account holds only 0.0333 BTC at $72,000, or 4% of the short's position value, and its equity reaches zero at $75,000. In the fully neutral $72,000 version, it reaches zero at exactly $72,000. The spot gains sit in a wallet the exchange cannot see, so the short can be liquidated in a rally while the overall portfolio has lost nothing. After liquidation, the investor is left long and unhedged.

The short's BTC position value also drifts, to 1.25 BTC at $48,000 and 0.83 BTC at $72,000, so re-matching the hedge to that number changes it. Real positions add further gaps: funding in bitcoin changes C over time, fees reduce it, the perpetual's mark can diverge from spot, and execution can differ from the model price. Losses are not necessarily capped at posted collateral either: in cross margin they can draw on other balances, and insurance-fund and auto-deleveraging rules vary by venue. Academic work on minimum-variance hedging with bitcoin inverse futures derives hedge ratios specific to this payoff structure.

Margin and liquidation in a coin-denominated account

Some principles hold across venues:

• Everything is measured in coin. Equity is the coin balance plus unrealized coin P&L, and position value is N / mark, so the margin ratio moves with both the trade and the coin's price.
• A long with coin collateral carries wrong-way risk. A falling price reduces both the position and the dollar value of the collateral, as Example 2 shows.
• A short with coin collateral is partly self-hedging in dollars. In the simplified model, a short collateralized with exactly N / P₀ BTC keeps equity equal to position value at every price. Any leverage beyond that lets the ratio fall as the price rises.
• Funding and fees move equity directly in coin. Fees and funding paid are debits; funding received is a credit.

Beyond these principles, risk engines differ: mark price construction, risk tiers, isolated versus cross versus portfolio margin, collateral haircuts and partial liquidation rules. Simplified thresholds are not exchange liquidation prices.

Linear vs inverse at a glance

Practical implications

For holders who measure wealth in bitcoin, inverse settlement keeps P&L in the asset they track, and coin collateral avoids converting into stablecoins. A coin-collateralized long, however, stacks on top of the bitcoin already held.

For hedging the dollar value of coin inventory, a dollar-neutral inverse hedge in the model above has to cover all relevant bitcoin, including collateral, not just the spot holding. A linear short with stablecoin collateral avoids the collateral term but needs stablecoins and leaves the hedge's gains in them. Both approaches share the fragmentation risk from Example 3 whenever the hedged asset sits outside the margin account.

Stablecoin settlement is not risk-free dollar settlement. Even Binance describes a USDT profit as only approximately its dollar value. Peg deviations, issuer risk and venue custody risk remain.

When comparing products, look at dollar exposure including collateral, the funding base, the liquidation path and contract specifications rather than headline leverage. Availability depends on jurisdiction and changes over time.

The Crypto Perpetuals & Basis Workbench models linear positions and coin-settled inverse perpetuals under defined combination restrictions, including certain spot-plus-inverse configurations. It does not model arbitrary mixed linear and inverse portfolios, unrestricted cross-currency collateral transfers, exchange portfolio margin or venue-specific liquidation engines.

Nothing here is a recommendation to trade any contract. The examples show mechanics under stated assumptions; real outcomes depend on execution, fees, funding, basis and venue rules.

The bottom line

Linear and inverse perpetuals can carry identical dollar exposure. They differ in where P&L is paid, how collateral moves, how funding is measured and what can be liquidated. Before trusting any hedge or return figure, write down the full position, including collateral, and value every component consistently at one price.