6.10.4 Dealer Capacity & Balance Sheets

Basic Prerequisites: 03 · The Search-and-Bargaining Model and Inventory Management & Quote Skewing.

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1. Intuition & Practical Objective

The DGP model assumes dealers face no capital limit - they can absorb any position and recycle it frictionlessly. Reality is the opposite: a dealer’s ability to bear risk is bounded by its equity, and when equity is depleted the dealer cannot intermediate. This page is the Duffie (2010) slow-moving capital story: a supply shock must be absorbed by the limited set of dealers with capital available right now; the price concession required is inversely proportional to that capital, and it reverses as capital arrives later.

The practical objective: understand why the immediate price impact of a shock scales as $1/\text{capital}$, why the recovery is gradual (search costs, time to raise capital), and how this connects to the 2008 crisis where dealer capital was depleted and arbitrage relationships (e.g. the CDS-bond basis) blew out.

1. Shocks hit a thin layer of capital. When someone must sell (a fund liquidation, a downgrade, a margin call), only the dealers currently willing and able to take the other side can absorb it. The smaller that set, the bigger the price concession.
2. The concession reverses. Dealers plan to “lay off” the risk as more capital arrives (new investors, recapitalisation, other dealers re-entering). So the initial impact is transitory: price dips, then recovers.
3. The reversal is slow and imperfect. Search frictions and capital-raising frictions (debt overhang, forced recapitalisation) delay it - that delay is what makes the immediate impact large.

The one-sentence result. “A supply shock of $q$ shares absorbed by a risk-averse dealer with capital $E$ requires a price concession $p\approx (\gamma {\sigma }^2/E)q$ - so impact scales as $1/E$ - and the concession decays as slow-moving capital arrives, producing the sharp-then-reversing price path of Duffie (2010).“

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2. Mathematical Ground Truth & Derivations

2.1 The capacity channel: impact $\propto q/E$

A dealer with capital $E$ and CARA (or mean-variance) risk aversion $\gamma$ absorbs $q$ shares of an asset with return variance ${\sigma }^2$. To hold the position it requires an expected excess return per share that compensates its risk:

$$\text{required return}=\gamma {\sigma }^2\frac{q}{E}.$$

Equivalently, the price must fall by the present-value of that concession. Taking a flat-discount approximation, the price concession (impact) is

$$\boxed{p(q,E)=\frac{\gamma {\sigma }^2}{E}q}$$

Two immediate consequences: impact is linear in the shock size $q$ (a small-capital dealer faces a linear impact, not the square-root law of a deep book), and impact is inversely proportional to capital $E$. Double the dealer’s capital and you halve the impact. The product $p\cdot E=\gamma {\sigma }^{2}q$ is the invariant: the dollar risk premium required is fixed by the position size and risk, so a thinner balance sheet must pay a proportionally larger price concession.

2.2 Slow-moving capital and the reversal

Capital does not arrive instantly. Let available intermediary capital evolve toward a long-run level $E_{\infty }$ with time constant $T$ (search for capital, recapitalisation delay):

$$E(t)=E_0+(E_{\infty }-E_0)(1-e^{-t/T}).$$

Because $p\propto 1/E$, the impact path is

$$p(t)=\frac{\gamma {\sigma }^{2}q}{E(t)}\underset{t\to \infty }{\to }\frac{\gamma {\sigma }^{2}q}{E_{\infty }}.$$

The defining signature is: sharp initial impact at $E_0$, then a gradual reversal to the small steady-state level as $E(t)\uparrow E_{\infty }$. Duffie (2010) documents exactly this shape in the CDS-bond basis, the 2005 General Motors/Ford index-arbitrage episode, catastrophe reinsurance, and convertible-bond arbitrage.

2.3 Why the reversal is incomplete / slow: capital-raising frictions

The recovery is delayed by frictions Duffie emphasises:

• Search frictions for trading counterparties (this folder’s core: $\lambda ,\rho$ small ⇒ slow).
• Capital-raising frictions: debt overhang (Myers 1984), limits to external equity, regulatory capital constraints.
• Fire-sale externalities: when many intermediaries are capital-constrained at once, they sell to the same thin set of buyers, so the price impact is amplified (a coordination failure), and losses feed back into capital.

Formally, when $K$ capital-constrained dealers all liquidate the same asset into a demand curve from the few unconstrained buyers, the aggregate impact is super-additive - the micro-foundation of fire-sale spirals.

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3. Computational Implementation - the impact/reversal path

We compute the capacity-channel impact across capital levels and trace the slow-capital reversal path. Stdlib + numpy. Ran and verified.

The invariant $p\cdot E=\gamma {\sigma }^{2}q=8.0\times 10^7$ holds exactly across every capital level - impact is purely a $1/E$ effect. The reversal path shows the Duffie (2010) signature: an immediate $4.00\%$ impact that decays to $0.71\%$ within a tenth of a year and to $0.09\%$ after two years, with a half-life of $T\ln 2=0.693$ years. The impact is front-loaded and transitory - the entire reason “slow-moving capital” explains both the crash and the subsequent recovery.

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4. Failure Modes & First-Principles Breakdowns

1. Capacity is procyclical and vanishes exactly when needed. Dealer capital $E$ is high in calm markets and collapses in stress - so the same shock produces a far larger impact in a crisis than in normal times. Marking impact with a constant $E$ is the central modelling error.
2. The reversal is not arbitrage-free to exploit. The “cheap” asset after a shock requires capital now to buy; if your capital is also constrained (or your funding is fragile), you cannot capture the reversal - the basis “arbitrage” is really a capital-intensive trade (Duffie 2010 documents its blowout in 2008).
3. Linear impact is an approximation. For very large $q$ or a deep electronic book the impact is often closer to square-root; the $1/E$ scaling is the capacity effect layered on top of the book-shape effect (see Market Impact & Depth).
4. Fire sales make the aggregate worse than the sum of parts. Independent liquidations share the same thin buyer set, so impact is super-additive - the coordination failure that turns a shock into a spiral.
5. Recapitalisation is slow and politically constrained. The 2008–09 episode required governments to force recapitalisation (SCAP, TARP); the delay is a first-order determinant of the size of the price distortion.

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5. References

• Duffie (2010), Presidential address: Asset price dynamics with slow-moving capital, Journal of Finance 65(4), 1237–1267
• Duffie, Gârleanu & Pedersen (2005), Over-the-counter markets, Econometrica 73(6)
• Duffie (2012), Dark Markets
• Myers (1984), The capital structure puzzle, Journal of Finance 39(3)
• Coval & Stafford (2007), Asset fire sales in equity markets, JFE 86(2)
• Hendershott & Menkveld (2014), Price pressures, JFE 114(3)

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6. Connected Graph Bridges

• Back: 03 · The Search-and-Bargaining Model
• Forward: 05 · Failure Modes & Practice · Index Hub
• Sibling: Inventory Failure Modes · Liquidity Risk & Funding