Margin, Leverage, and the Spiral

Leverage is usually introduced as a multiplier: borrow to hold a larger position, and both gains and losses scale up. That description is arithmetically correct and it omits the part that causes failures.

Borrowing introduces a counterparty with rights. The lender requires collateral, revalues it continuously, and may demand more at any time. If the demand is not met, they close the position themselves.

That changes the nature of the risk rather than its size. An unleveraged holder facing a 40% decline has a loss and a decision. A leveraged holder facing the same decline may have no decision, because the position was closed at the bottom by someone else acting under a contract.

So the question a leveraged book must answer is not how much it might lose. It is at what price it stops being allowed to hold the position, which is a different quantity with a different sensitivity.

Two distinct payments are both called margin, and confusing them is a common source of error.

Initial margin is collateral posted when the position opens. It is sized to cover a plausible adverse move over the time it would take to close the position out, so it scales with volatility and with expected liquidation time. It is a buffer, not a payment: it comes back when the position closes.

Variation margin is the daily settlement of gains and losses. If the position loses, cash moves to the counterparty. This is not a buffer and it does not come back. It is the loss, realised in cash, every day, whether or not the position is eventually profitable.

The distinction produces the classic failure. A position that is correct over six months but loses heavily in month one generates real cash outflows in month one. A trader who is right and cannot fund the interim variation margin is liquidated before being proved right.

Solvency and liquidity are separate conditions, and the second one binds first.

Given a maintenance margin requirement, the price at which the position is closed is arithmetic, and it is worth computing before entering rather than during.

def liquidation_price(entry, equity, notional, maintenance=0.25):
    """Long position. Returns the price at which equity falls to the
    maintenance fraction of position value."""
    units = notional / entry
    # equity(P) = units*P - debt ; debt = notional - equity
    debt = notional - equity
    return debt / (units * (1 - maintenance))

liquidation_price(100.0, equity=50_000, notional=100_000)   # 66.67
liquidation_price(100.0, equity=25_000, notional=100_000)   # 100.00
liquidation_price(100.0, equity=20_000, notional=100_000)   # 106.67

Read the three results together. At 2x leverage the position survives a 33% decline. At 4x it is at the liquidation threshold on day one. At 5x the required price is above the entry price, meaning the position cannot be opened at all under that requirement.

The relationship is not linear. Doubling leverage does not halve the survivable move; it collapses it toward zero, and the collapse accelerates.

The calculation above treated the maintenance requirement as fixed. It is not, and the way it moves is the heart of the problem.

Margin requirements are set from estimated volatility and estimated liquidation cost. Both rise in a stress episode. So the requirement increases at the same moment the position is losing.

The two effects compound in the same direction. Equity is falling because of the loss, and the required equity is rising because of the volatility. A position comfortably within limits can breach them without any further price movement, purely because the requirement was raised.

This is procyclical margining, and it is the mechanism connecting the previous lesson's estimated-distribution problem to actual forced selling. The risk model that understated danger in calm conditions was also setting the margin requirement, and its correction arrives as a cash demand rather than as a revised report.

Anyone planning leverage on current requirements is planning on the calm-period value of a number that moves against them precisely when it matters.

Markus Brunnermeier and Lasse Heje Pedersen modelled this in "Market Liquidity and Funding Liquidity", Review of Financial Studies, volume 22, issue 6, 2009, pages 2201 to 2238. Their result is that under certain conditions margins are destabilising, and market liquidity and funding liquidity become mutually reinforcing, producing liquidity spirals.

Two distinct things are being connected. Market liquidity is how easily an asset trades. Funding liquidity is how easily a trader raises cash. Ordinarily these look like separate concerns.

The loop ties them. Falling prices tighten funding, which forces selling, which worsens market liquidity, which raises the cost of the next liquidation, which tightens funding further.

The branch to the right matters as much as the main loop. Market makers are leveraged too, so the same margin call that forces a fund to sell also reduces the capacity of the participants who would normally absorb that selling. Supply of liquidity falls at the exact moment demand for it spikes.

The spiral has an implication that no single-portfolio risk model can produce, because it depends on positions the model cannot see.

If many participants hold the same position with similar leverage, they receive margin calls simultaneously and sell the same asset at the same time. The price impact each of them causes is inflicted on all the others, and each one's forced selling triggers the next.

This is why concentration in a crowded position is more dangerous than the position's own volatility suggests. The relevant question is not how volatile the asset has been but who else owns it, at what leverage, and what would force them to sell.

It also explains a pattern that looks paradoxical from the outside: positions that appear low risk by historical measures can be the most dangerous ones. Low measured volatility attracts leverage, leverage attracts more participants, and the crowd is the risk. The measured calm is not evidence of safety; it is part of the cause.

A risk model computed from your own book alone is structurally blind to this.

Since the loop cannot be avoided, the practical question is what makes a book able to sit through one.

Unused borrowing capacity. Leverage held below the maximum available is not idle capital; it is the ability to meet a call without selling. This is the single most effective defence and the most expensive in quiet periods.

Cash that is genuinely cash. Collateral that must itself be sold to be useful is not a buffer, because it is worth least at the moment it is needed.

Sizing to the liquidation price, not the expected loss. The relevant question is not what happens on a typical bad day but at what price the decision leaves your hands.

Diversity of funding. A single lender able to raise requirements unilaterally is a concentration even when the positions are not.

Each of these costs money continuously and pays only rarely, which is why they erode during long calm periods. That erosion is not usually a decision anyone makes. It is the cumulative result of many reasonable-looking optimisations, each of which improves returns in the environment that has been observed.

The second row is the compressed version of the whole lesson. Leverage transfers the choice of when to exit from the holder to the lender, and the lender exercises it under a rule that fires in the worst conditions.

Everything else follows. Being right about the direction is insufficient if the path passes through your liquidation price. The relevant risk measure for a leveraged book is not a loss distribution but the distance to that price, and how that distance behaves when volatility rises.

Which leaves the question of how large a position should be in the first place. That is the last lesson.