Borrowing, equity and collateral repair

What you will learn

Derive financed owner returns and a maintenance price boundary, prove the sale-and-repayment equity invariant, and certify minimal deleveraging versus insolvency.

Prerequisites: Target returns and the feasible frontier · Portfolio weights, drift and rebalancing

Account for the liability before measuring performance

The previous frontier forbade negative cash. Allowing borrowing changes more than the list of permitted weights: it introduces a liability with its own growth clock and a collateral condition. Start with owner equity E0=100 fictional chips. Borrow B chips and spend the combined amount A0=100+B on one divisible asset. The balance sheet reads assets minus debt equals equity. Borrowing raises assets and debt by the same amount, so it creates purchasing power without creating owner wealth. Define initial asset leverage L=A0/E0. Borrowing 100 gives L=2; it does not give initial wealth 200.

Assume a single stress horizon, a known asset simple return r, and a known simple financing rate i for precisely that horizon. Interest is accrued into debt before the collateral test. There are no interim payments, dividends, taxes, fees, bid-ask costs, price impact, outside contributions, short positions or further interest during repair. The asset can be sold fractionally at the stress mark, and every sale chip immediately repays debt. The fictional lender accepts equity at least h times remaining asset value, with 0<h<1. These are stipulated teaching terms, not current account eligibility, regulatory requirements or a forecast. No probabilities are assigned to the stress cases.

Derive the financed return and a price boundary

At the horizon, asset value is A=(100+B)(1+r) and debt is D=B(1+i). Equity is E=A−D=100+(100+B)r−Bi. Dividing the change by initial owner equity gives RE=(E−100)/100=Lr−(L−1)i. This return uses the owner's original capital as denominator, rather than the larger asset purchase. At B=100, i=0.10 and r=0.20, equity is 240−110=130, giving 30% rather than 40%. The missing ten percentage points are the ten-chip financing charge relative to the initial hundred. At r=−0.40 the same formula gives −90%, leaving equity ten.

The collateral test is E≥hA, equivalently A−D≥hA or (1−h)A≥D. For the positive asset values in this lesson, divide by (1−h)A0 to obtain r≥D/((1−h)A0)−1. This is a price-return boundary at a declared clock, not a probability of a margin call. With B=100, i=0.10 and h=1/4, it equals 110/150−1=−4/15, about −26.6667%. Equality satisfies the rule. A forty-percent decline violates it despite leaving positive equity. Raising h or increasing accrued debt raises this boundary. A solvent account can therefore fail collateral requirements long before equity reaches zero.

Separate the equity shortfall from the asset sale

At the forty-percent decline, A=120 and D=110, so E=10. Required equity is hA=30, leaving a collateral shortfall of 20. It is tempting to sell 20 of assets and call the shortfall repaired. But selling assets to repay debt does not add new owner equity. For sale proceeds x with 0≤x≤min(A,D), post-sale assets are A−x, debt is D−x, and equity is (A−x)−(D−x)=A−D=E. The sale improves the equity fraction by shrinking its denominator, not by reversing the price loss. This cancellation is the central invariant of the repair.

Require E≥h(A−x). Rearrangement gives x≥A−E/h=(hA−E)/h. For positive equity, the least sale is x*=max(0,A−E/h). In our example x*=120−10/(1/4)=80. Assets become 40, debt 30 and equity stays ten; ten divided by forty is exactly one quarter. Selling only the 20-chip shortfall leaves assets 100 and debt 90, so the fraction is only ten percent. An external cash injection repaying debt would follow different algebra because it increases equity; it is deliberately unavailable here. Never substitute a deposit amount for a sale amount without reconstructing both balances.

Prove minimality and handle exhausted equity

The inequality itself is a continuous minimality certificate: every x below 80 leaves h(A−x)>E, while x=80 meets equality. Independently test neighboring ten-chip moves. At sale 70, assets are 50, debt 40, equity ten and the fraction is 20%, which fails. At sale 90, assets are 30, debt 20 and the fraction is one third, which passes but sells more than necessary. Feasible does not imply minimal. In a positive-equity breach, x*=A−E/h is below A and at most D, since D−x*=E(1/h−1)≥0. Thus debt repayment and available assets do not invalidate the candidate under our assumptions.

When E<0, sale-and-repayment cannot repair the deficit: equity remains negative and even selling everything leaves unpaid debt D−A. At B=100, i=0.10 and r=−0.60, assets are 80, debt 110 and the owner owes 30 after full liquidation, a −130% equity return. Treat this as insolvent in the workshop, enter no sale certificate and explain the residual liability. If E=0, any positive remaining asset value fails a positive maintenance requirement. Selling all assets exactly repays debt and closes the account with zero equity; the ratio 0/0 is undefined, not zero percent and not a surviving compliant position. Our selectable stresses avoid this knife edge, but the derivation must still identify it.

Construct a repair and retain a comparison witness

Use the workshop's debt, horizon rate, stress and maintenance controls to rebuild the balance sheet. Transfer ten-chip sale packets from the stressed asset into debt repayment. The displayed post-sale rows let you check the equity invariant after each move. Supply accrued debt, pre-sale equity, minimum sale and the pre-sale classification safe, breach or insolvent. The workshop does not automatically construct the repair for you. In an insolvent case, leave the minimum-sale entry blank and retain zero sale packets; the certificate concerns impossibility of repair, not a claim that liquidation has no value.

Begin with the default breach and compare a sale of 20 against 80. Save the verified repair, then raise maintenance from one quarter to one half. The same assets and debt now require sale 100, leaving assets 20 and debt ten. The earlier witness remains readable but cannot certify the edited model. Remove financing interest while keeping the forty-percent decline: debt becomes 100, equity 20 and the quarter-maintenance minimum sale falls to 40. Finally select the sixty-percent decline and certify insolvency, then remove borrowing to obtain an unlevered safe account. Model edits clear sale packets and answers; answer or packet edits clear success feedback. These comparisons isolate financing and collateral effects without inventing independent trials, expected returns, a trading advantage or real execution access.

Hands-on: Build a collateral repair

Vary debt, interest, stress and maintenance, then construct a minimal sale-and-repayment certificate or prove insolvency.

Common mistakes

• Dividing owner gain by gross assets rather than initial owner equity.
• Forgetting accrued interest before testing collateral.
• Selling the equity shortfall instead of dividing it by the maintenance fraction.
• Counting sale proceeds as new equity while also reducing debt.
• Calling a solvent collateral breach insolvency, or assigning a percentage to 0/0.

Keep these ideas

• Equity is assets minus accrued debt; RE=Lr−(L−1)i.
• Maintenance is E≥hA, giving r≥D/((1−h)A0)−1.
• Sale plus equal debt repayment preserves E; for E>0 minimum sale is max(0,A−E/h).
• Negative equity cannot be repaired by internal sales; zero equity only permits a full close under these assumptions.

Further reading

• SEC Investor Tips: Margin, borrowing money to pay for stocks ↗

  Read Understand How Margin Works, Recognize the Risks, Amount You Need After You Trade and Margin Calls. Used for collateral, assets-minus-debt equity and forced-sale concepts only. Historical page is not used as current regulatory guidance. Fictional terms, stress cases, invariant, minimum-sale derivation and diagnostics are original; no source numerical example or video reused. Checked 2026-10-25.

• SEC Investor Bulletin: Interested in Margin? Understand Interest (2022) ↗

  Read Interest Charges and questions on accrual and reducing loan balance. Supports explicit financing-cost and clock assumptions only. Source netting examples are not reused; our simple horizon rate is stipulated rather than a claim about any provider. No quote, credential or video review claimed. Checked 2026-10-25.