The Verification Venue · pointed at the arithmetic underneath a banking panic
The Bank That Dies at Three Percent
When a bank fails, the headline is always the size of the loss. Size was never the question. A loss only matters as a fraction of assets, and the fraction that kills a bank is exactly one divided by its leverage. At 10 to 1 that is 10 percent. At 30 to 1 it is 3.33 percent, a shade over three. This page hands you the balance sheet; push it over yourself.
Below is a stylised bank with $100 billion of assets. Its equity, the shareholders' money that absorbs losses, is set by one number: leverage, assets divided by equity. Drag the leverage, then drag the size of the loss. Watch the equity slice of the balance sheet thin, hit zero, and go through it.
Five stylised balance sheets, same $100 bn of assets: L = 10
These are archetypes drawn for shape, not portraits of any named institution; a tile only sets the leverage slider. ↓
Equity after the loss
$5.00 bn
status: SOLVENT
Distance to ruin
5.0 pp
$5.00 bn of further losses reaches the threshold
Drag it to 30 and watch the fatal loss shrink to about one thirtieth of assets. Drag it to 5 and the bank shrugs off almost anything short of a fifth.
Push it past the threshold readout and the status flips from SOLVENT to INSOLVENT; keep going and the shortfall bar grows.
One definition, used everywhere on this page: leverage is total assets divided by total equity. Not Tier 1, not tangible common equity, not whichever adjusted figure makes a quoted ratio look sturdier. Numbers you meet elsewhere vary in the numerator and the denominator, sometimes a lot, which is why this page defines the ratio once and never quietly swaps it.
Write A for assets, L for leverage, f for the fraction of assets lost. Equity afterwards is A(1 − f) minus the liabilities, and the liabilities are A(1 − 1/L). Everything cancels to one line:
Set the leverage to 10 and the fatal loss is exactly one tenth of assets. Set it to 30 and the fatal loss is 1/30 of assets, 3.33 percent, which is where the title comes from. One boundary stated exactly: at f = 1/L equity is exactly zero, and that zero is the ruin point; for every f greater than 1/L equity is negative, and for every f less than 1/L it is positive. The threshold is not a rule of thumb or an empirical regularity. It is the whole of the ruin arithmetic: everything on this page about when the bank dies follows from that one line. The risk-weighting section further down is different arithmetic, built on stipulated policy inputs rather than this identity, and it is labelled as such where it appears. The check panel below recomputes the threshold two independent ways to machine precision.
Now the sophisticated dismissal: leverage is a blunt instrument, regulators do not look at raw size, they look at risk-weighted assets, because a dollar of government debt is not a dollar of speculative equity. That is true, and worth taking seriously, which is why this page does not argue with it. It builds the second balance sheet and lets you produce the complaint yourself.
| bucket | share of book at the start | risk weight (illustrative) |
|---|---|---|
| Cash and central-bank claims | 5% | 0% |
| Domestic sovereign debt | 15% | 0% |
| Senior secured mortgage | 40% | 35% |
| Corporate loan | 35% | 100% |
| Unrated equity holding | 5% | 150% |
These weights are a stylised set, stipulated for this page to make the mechanism visible. They are not any regulator's schedule, they follow no named framework version, and this page is not describing any jurisdiction's current rule.
Risk-weighted assets
$56.50 bn
sum of exposure × weight over the five buckets
Required capital (illustrative ratio)
$4.52 bn
ratio × risk-weighted assets
Capital freed vs zero shift
+$0.00 bn
held constant, this surplus sits idle against an unchanged risk
Loss that ruins this bank
10.00%
set by leverage alone; no composition change moves it
Total size never changes; only the mix does. Fully right, the whole corporate book is sovereign debt. Watch required capital fall while the threshold readout stays frozen.
A round, stipulated number chosen for legibility. No jurisdiction's actual ratio appears on this page.
Drag the shift to its full +35 points and the entire corporate book has become sovereign debt. Required capital drops by $2.80 bn, capital the bank no longer needs to hold against this portfolio. Now look at the fourth readout: the loss that would exhaust equity has not moved by a single basis point, because equity and assets have not moved. That coexistence, capital freed and ruin threshold frozen, is the quantitative form of the complaint about risk weighting. It is also, deliberately, all this page claims. The arithmetic here does not say risk weighting caused any particular crisis, and it would be dishonest decoration to imply it.
One honest caveat, named where you can act on it: the freed capital is only free while it stays on the balance sheet. Pay it out as dividends or buybacks and equity falls, leverage rises, and the ruin threshold drops with it. This page holds equity constant throughout, and the readouts show the retained-surplus case. Which of those worlds you are in is a policy choice, not an accounting one.
The check: every number recomputed in front of you
The first table covers seven leverages: 2, 5, 10, 12.5, 20, 30 and 40. Each row is shown twice: once as the closed form 1/L, once by bisection on the equity function, a root-find that never divides 1 by L. The rows ship pre-rendered so the table exists before any script runs, and the page's script recomputes every cell on load from the same formulas, replacing what you see and flagging any cell that drifts; the columns agree to floating-point precision or the row shows a cross. Below, the risk-weighted assets of your current portfolio are summed three different ways; addition is associative, so any disagreement is a bug.
| leverage L | 1/L closed | bisection | |diff| | ok |
|---|---|---|---|---|
| 2 | 0.5000000000 | 0.5000000000 | live | ✓ on load |
| 5 | 0.2000000000 | 0.2000000000 | live | ✓ on load |
| 10 | 0.1000000000 | 0.1000000000 | live | ✓ on load |
| 12.5 | 0.0800000000 | 0.0800000000 | live | ✓ on load |
| 20 | 0.0500000000 | 0.0500000000 | live | ✓ on load |
| 30 | 0.0333333333 | 0.0333333333 | live | ✓ on load |
| 40 | 0.0250000000 | 0.0250000000 | live | ✓ on load |
What is exact here: the algebra, the root-find agreement, and the conservation identity. What is stipulated: the $100 bn size, the five weights, the ratio, all labelled as stipulations where they appear. Run it yourself: node research/the-bank-that-dies-at-three-percent/verify-the-bank-that-dies-at-three-percent.mjs
What's idealised here, and what's exactly true
Exactly true. The identity E_after = A(1/L − f) is two lines of algebra, not a model. Equity is exactly zero at f = 1/L, positive below it and negative above it; that sign structure is the whole claim. The bisection column finds the same root by repeated halving without ever forming 1/L as the answer, and agrees with the closed form to floating-point precision. The conservation check is associativity of addition: regroup the same multiplications and the sum cannot change.
Idealised. The balance sheet is static, and the loss is a single instantaneous write-down applied pro rata across all assets. There are no off-balance-sheet exposures, no derivatives, no contingent calls, no taxes, no interim income, no funding run. The $100 bn size, the five weights, and the 8 percent ratio are round stipulations chosen for legibility. Equity means total accounting equity; real-world quotes often substitute narrower measures, which is exactly the definitional drift the mechanism paragraph warns about.
Representative, not universal. Reported leverage varies widely depending on the numerator and denominator chosen, and the weights vary across frameworks and jurisdictions. What survives every variation is the direction: raise L and the threshold f* = 1/L falls, always, at exactly the reciprocal rate. The page's claim lives in that invariance, not in any particular constant.