4.6.4 Margin, Funding Spirals & the Brunnermeier–Pedersen Mechanism

Basic Prerequisites: 03 · Liquidation Cost & L-VaR and VaR & Expected Shortfall.

---

1. Intuition & Practical Objective

This is the engine room of the folder. The two spirals of Brunnermeier & Pedersen (2009) are simple enough to write as recurrences and simulate:

• Loss spiral. Falling prices reduce equity $N$, which reduces the maximum affordable position $N/m$, which forces selling $P-N/m$, which pushes prices down further. The numerator shrinks.
• Margin spiral. The same stress raises volatility, the broker hikes the haircut $m$, which also reduces the maximum position $N/m$. The denominator grows.

They are usually described as one “liquidity spiral,” but they are two distinct positive-feedback channels and they fire simultaneously - which is why the effect is multiplicative rather than additive. A fund that could survive a 5% loss at constant margin can be wiped out by a 3% loss that drags the margin up behind it.

The objective of this page. Turn the prose of the Brunnermeier–Pedersen paper into a runnable recurrence that shows, round by round, how a small exogenous shock becomes a large forced liquidation - and identify the three parameters (leverage, margin sensitivity to vol, market-impact coefficient) that decide whether the process converges or runs away.

---

2. Mathematical Ground Truth & Derivations

2.1 The margin constraint and the two-armed squeeze. A financier holding position value $P$ with equity $N$ and margin (haircut) $m$ faces

$$P\le \frac{N}{m}⟺\text{required deleveraging}={\left(P-\frac{N}{m}\right)}^{+}.$$

In Brunnermeier–Pedersen the speculator’s demand is $P=\max \left\{N/m\text{(funding-limited)},y\text{(unconstrained)}\right\}$, so when the funding constraint binds, every shock propagates. Before a shock, $P=N/m$; after a shock that changes both $N$ and $m$,

$$\Delta P_{\text{forced}}=\frac{N}{m}-\frac{N^{\prime }}{m^{\prime }},N^{\prime }=N-\underset{\text{loss spiral}}{\underbrace{(\text{loss})}},m^{\prime }=m+\underset{\text{margin spiral}}{\underbrace{\beta \Delta \text{Vol}}}.$$

Both terms in $\frac{N^{\prime }}{m^{\prime }}=\frac{N-\text{loss}}{m+\beta \Delta \text{Vol}}$ move for the worse: the fraction decreases because its numerator falls and its denominator rises.

2.2 Margin set from risk (the margin spiral’s driver). Haircuts are set from the collateral’s VaR over the margin period of risk,

$$m=z_{\alpha }{\sigma }_{\text{MPOR}},{\sigma }_{\text{MPOR}}=\sigma \sqrt{\Delta t_{\text{MPOR}}},$$

so $m$ rises with $\sigma$ and with the lengthening liquidation delay $\Delta t_{\text{MPOR}}$ of the collateral - and that delay is exactly market illiquidity. This is the coupling that makes the two liquidities one system: market illiquidity lengthens MPOR, MPOR raises the haircut, the haircut forces sales, the sales worsen market illiquidity.

2.3 Loss spiral as an amplification factor. If forced sales of size $S$ move the price by $\kappa S$ (linear impact, $\kappa =\text{impact coefficient}/ADV$), and a fraction $L=P/N$ of the resulting equity loss is re-deleveraged, the total liquidation is the geometric series

$$S_{\text{total}}=S_0\sum _{k\ge 0}(L\kappa {)}^k=\frac{S_0}{1-L\kappa },L\kappa <1.$$

The amplification factor is $1/(1-L\kappa )$. It is benign at $L\kappa =0.1$ ($\times 1.1$) and explosive as $L\kappa \to 1$: at $L\kappa =0.5$ the initial $10M sale becomes $20M; at L\kappa\ge1$ the series diverges - deleveraging cannot keep up with the price impact it creates. This is the formal statement of “the spiral has no fixed point.”

2.4 The loss spiral across agents (fire-sale externality preview). When many agents share the same collateral and margin rules, each agent’s $\kappa$ depends on the aggregate sale $S_{\text{agg}}$, not its own. Private optimisation uses $\kappa S_i$; the social cost uses $\kappa S_{\text{agg}}$. Because $S_{\text{agg}}>S_i$, the private cost is systematically too low - a pecuniary externality (see 05 · Failure Modes).

2.5 The regulatory layer (BCBS). Funding-liquidity risk is now capital-adjacent via two ratios:

$$\text{LCR}=\frac{\text{HQLA}}{\text{Net cash outflows over 30 days}}\ge 100\%\text{(BCBS 2013, d238)},$$ $$\text{NSFR}=\frac{\text{Available stable funding}}{\text{Required stable funding}}\ge 100\%\text{(BCBS 2014, d295)}.$$

The LCR forces enough high-quality liquid assets to survive 30 days of stress outflows - a direct defence against the funding spiral. Its procyclicality is the known cost: the required stock of HQLA rises exactly when everyone wants to hold it, and the LCR’s run-off assumptions can dictate the sale of the very assets under stress (see 06 · Advanced Extensions).

---

3. Computational Implementation - a deterministic margin-spiral simulation

We run the recurrence of §2.1–2.3 on a $50M book funded at 5× leverage. Each round: the broker hikes the haircut (margin spiral), the constraint binds, the shortfall is sold, and the sale’s market impact feeds back as a mark-to-market loss (loss spiral). Stdlib only; fully deterministic.

Read it carefully. A 3% price shock ($1.5M of loss on a $10M equity base) triggers the sale of 74.6% of a $50M book - a $37.3M liquidation - not because the fund’s view changed, but because the haircut walked from 20% to 60% while the equity base eroded. The first round alone dumps $14.5M (29% of the book) and moves the price a further 3.6%, nearly re-creating the initial shock out of the fund’s own selling. This is the margin spiral doing the damage: with zero impact and a constant haircut the loss spiral alone would still force a $6.0M sale (the 3% shock pushes leverage to 5.71× past the 5× limit), settling at $42.5M position / $8.5M equity / 5.0× - a single forced sale, not a cascade.

A lever to pull. The three parameters that decide convergence are leverage $L$, haircut sensitivity $\text{d}m/\text{d}\text{Vol}$, and $\kappa$ (impact). Setting impact_coef=0 recovers the pure loss spiral; raising haircut_step to 0.15 collapses the book from $48.5M to $8.1M in three rounds while equity stabilises near $4.8M (the m_cap=0.60 bounds the spiral; pick parameters past the explosive boundary to get a true divergence). The explosive boundary is exactly §2.3’s $L\kappa \to 1$.

---

4. Failure Modes & First-Principles Breakdowns

1. “Two spirals” treated as one. Confusing the loss spiral (numerator) with the margin spiral (denominator) hides the fact that a policy fixing one - e.g., a margin holiday - still leaves the other. Brunnermeier–Pedersen stress that the interaction is what makes the funding constraint bind so hard.
2. Endogenous margin assumed exogenous. If you model the haircut as a constant you delete the margin spiral by construction and will persistently under-forecast forced sales. The correct object is $m(\sigma ,\text{MPOR})$ with $\sigma$ and MPOR both rising in stress.
3. Ignoring the price impact of your own liquidations. The round-1 sale of $14.5M moved the price 3.6%; a model that liquidates at the pre-sale price understates the equity burn by the whole impact term and can declare “survived” when the fund is gone.
4. Aggregation failure. At the system level, simultaneous deleveraging means each fund’s $\kappa$ is set by the aggregate sale. Firm-level L-VaR (03 · §2.2) systematically understates the cost each fund bears in a crowd.
5. The LCR’s own procyclicality. Requiring HQLA to cover 30-day outflows is protective at the firm level, but when many banks must all meet the ratio in the same stress, the demand for HQLA rises and their prices fall - the regulator’s buffer becomes a buyer’s strike. The NSFR’s structural maturity-matching can likewise push funding into exactly the short-tenor markets that freeze first.
6. MPOR lengthening is the hidden multiplier. Market illiquidity raises $\Delta t_{\text{MPOR}}$, which raises $m$ via $\sqrt{\Delta t_{\text{MPOR}}}$ - a channel a “market-risk-only” margin model misses, because it lives on the funding side while being driven by the market side.

---

5. References

• Brunnermeier, M. & Pedersen, L.H. - Market Liquidity and Funding Liquidity, RFS 22(6):2201–2238 (2
• Brunnermeier, M. - Deciphering the Liquidity and Credit Crunch 2007–2008, JEP 23(1):77–100 (2
• Foucault, Pagano & Röell - Market Liquidity (2013)
• Shleifer, A. & Vishny, R. - The Limits of Arbitrage, JF 52(1):35–55 (1997)
• BCBS - Basel III: The LCR and Liquidity Risk Monitoring Tools (2013, d238); Basel III: The NSFR (2014, d295). The regulatory funding-liquidity layer of
• Hull - Risk Management and Financial Institutions, liquidity-risk and Basel chapters

---

6. Connected Graph Bridges

• Back: 03 · Liquidation Cost & L-VaR · Index Hub
• Forward: 05 · Failure Modes · 06 · Advanced Extensions (Liquidity Spirals & Systemic Risk)
• Siblings: Stress Testing & Scenario Analysis · Credit Risk & the Merton Model
• Portfolio bridge: Transaction Costs & Turnover Constraints · Flat note: Liquidity Risk & Margin Spirals