Homemade Leverage and Arbitrage

Why Investors Can Recreate Leverage

A company can change how it finances its operating assets by issuing debt or equity. Under the perfect-market assumptions used for the no-tax capital structure model, an investor can also change the leverage of a personal portfolio: borrow to increase exposure to an unlevered company, or lend alongside levered equity to remove the effect of corporate debt. The key is to match both the initial cost and the cash flows in every modeled state.

This equivalence is useful for more than portfolio construction. If two portfolios deliver the same future cash flows but have different prices, an investor can buy the cheaper portfolio and sell the more expensive one. That trade is an arbitrage under the model assumptions. The analysis below focuses on the construction and price comparison, not on how a company chooses its debt policy.

Definition: Homemade leverage is leverage created by an investor’s own borrowing or lending rather than by changing the company’s financing. A portfolio is a replicating portfolio when it has the same cash flows in every relevant state and at every relevant date as the position it is intended to reproduce.

Use the following notation. Let $V_U$ be the market value of an unlevered company’s equity, which is also the value of its operating assets. Let $V_L$ be the total market value of a levered company, $E_L$ its equity value, and $D$ the market value of its debt. Let $Z$ denote the operating assets’ future cash flow. For the simple one-period illustrations, debt has a specified payoff at the future date, and it is paid in full in every modeled state. Thus the levered equity payoff is the operating cash flow less the debt payoff.

Assume no corporate or personal taxes, transaction costs, or differences in borrowing and lending terms; securities are fairly traded and divisible; investors can borrow and lend at the same rate and terms as the company; and the relevant debt payoff is met in every modeled state. The last condition makes the debt claim’s payoff matchable by personal borrowing or lending. If default, limited liability, taxes, or borrowing constraints matter, the simple replication may fail and the conclusions need not apply.

Replicating Levered Equity with Personal Borrowing

Suppose an investor wants a fraction $\alpha$ of the operating assets’ cash flow, but the company itself is unlevered. The investor can buy $\alpha$ of the unlevered equity and borrow personally an amount equal to $\alpha$ times the market value of the company’s debt in the comparable levered financing. The personal loan must have the same payment dates and payoff terms as that corporate debt.

The investor’s initial net cost is the purchase of unlevered equity less the personal borrowing. At the future date, the investor receives a fraction of the operating cash flow and pays the matching fraction of the debt payoff. Those are the cash flows of the same fraction of levered equity.

\begin{aligned} \text{Initial cost} &= \alpha V_U-\alpha D,\\ \text{Future payoff} &= \alpha Z-\alpha(\text{debt payoff}),\\ \text{Levered-equity payoff} &= \alpha\bigl(Z-\text{debt payoff}\bigr). \end{aligned}

The payoff expressions are equal because multiplying the company’s net equity payoff by $\alpha$ gives the investor’s net payoff. Under the no-tax value relationship $V_U=E_L+D$, the initial cost also equals $\alpha E_L$. Thus the homemade-leverage portfolio matches both the cost and the payoff of owning fraction $\alpha$ of the levered equity.

Worked Example: Borrowing to Match Levered Equity

An unlevered company’s equity is worth $140. A comparable levered company has debt worth $40 and equity worth $100. The debt is due to pay $42 in one year, and the borrowing rate is 5% for the year, so a $40 loan requires a $42 repayment. An investor wants the cash flows of 15% of the levered equity. The operating assets produce a one-year cash flow $Z$, and the debt is paid in full in every modeled state. Use personal borrowing to construct the position and verify its cost and payoff. Method: scale both the equity purchase and debt financing by the target ownership fraction. This avoids treating the borrowed amount as an extra investment cost.

Step 1: Scale the unlevered equity holding. Buy 15% of the unlevered company. Its cost is $0.15(140)=\$21$, and its future cash flow is $0.15Z$.

Step 2: Scale the personal loan. Borrow 15% of the debt’s current value: $0.15(40)=\$6$. The investor receives $6 now and repays $0.15(42)=\$6.30$ in one year.

Step 3: Calculate net initial cost. The equity purchase costs $21, while the loan supplies $6. Therefore, net cost is $21-6=\$15$.

Step 4: Compare future payoffs. The homemade position pays $0.15Z-\$6.30$. The levered company’s equity pays $Z-\$42$, so 15% of that equity pays $0.15(Z-42)=0.15Z-\$6.30$. The cash flows match.

Check. Fifteen percent of the levered equity’s market value is $0.15(100)=\$15$, equal to the homemade position’s net cost. The construction avoids the mistake of subtracting the loan from future cash flow without also subtracting the matching scaled debt payoff from levered equity.

Removing Leverage by Lending

The reverse construction starts with levered equity. To reproduce a fraction $\alpha$ of unlevered equity, buy fraction $\alpha$ of levered equity and lend an amount equal to fraction $\alpha$ of the corporate debt’s current value. The lending position earns the debt payoff on the same schedule as the debt claim.

The levered equity contributes $\alpha$ times the operating cash flow less the debt payoff. The personal loan contributes back the debt payoff. Adding them leaves precisely $\alpha$ times the operating cash flow. This is the cash flow from fraction $\alpha$ of unlevered equity.

\begin{aligned} \text{Initial cost} &= \alpha E_L+\alpha D,\\ \text{Future payoff} &= \alpha(Z-\text{debt payoff})+\alpha(\text{debt payoff})\\ &=\alpha Z. \end{aligned}

Worked Example: Lending to Reproduce Unlevered Equity

A levered company has equity worth $224 and debt worth $56. Its unlevered counterpart is worth $280. The debt’s payoff in one year is $70, and it is paid in full in every modeled state. An investor wants the cash flows of 25% of the unlevered company. Determine the levered-security portfolio, its cost, and its payoff when the operating assets’ one-year cash flow is $400. Method: buy the same fraction of levered equity and lend the same fraction of the debt value. This avoids the common error of buying levered equity alone and overlooking the debt claim.

Step 1: Buy 25% of levered equity. The cost is $0.25(224)=\$56$. With a $400 operating cash flow, the company’s equity payoff is $400-70=\$330$, so the investor receives $0.25(330)=\$82.50$.

Step 2: Lend 25% of the debt’s current value. The loan costs $0.25(56)=\$14$. It pays $0.25(70)=\$17.50$ in one year.

Step 3: Add the initial costs and future receipts. Initial cost is $56+14=\$70$. The future payoff is $82.50+17.50=\$100$.

Check. A 25% ownership interest in unlevered equity costs $0.25(280)=\$70$ and receives $0.25(400)=\$100$. Both the initial cost and future payoff agree.

Why Equal Cash Flows Imply Equal Values

The central pricing argument is a direct consequence of the no-arbitrage principle: two portfolios with identical cash flows in every state and at every date must have the same market value in a perfect market. Otherwise, buying the cheaper one and selling the dearer one produces a positive initial cash receipt with no net future obligation.

Apply that reasoning to the unlevered company and the combined securities of the levered company. Unlevered equity pays the operating cash flow $Z$. Levered equity pays $Z$ less the debt payoff, while debt holders receive the debt payoff. Together, levered equity and debt pay $Z$. Their total payoff therefore matches unlevered equity in every modeled state.

Theorem: Under the stated no-tax, perfect-market assumptions, and when debt payoffs are matched in every modeled state, equal operating-asset cash flows imply $V_U=E_L+D=V_L$. If the prices fail this equality, buying the cheaper cash-flow claim and selling the more expensive equivalent claim creates an arbitrage in the model.

The justification has two parts. First, at the future date, the levered equity and debt payoffs add to $Z$, the same payoff as unlevered equity. Second, if their current prices differed from the price of unlevered equity, the offsetting future cash flows would leave a strictly positive initial price difference. A positive receipt now combined with zero future net cash flows is an arbitrage. Therefore, consistent prices must satisfy the equality.

Write the cash flows.
List unlevered equity, levered equity, and debt payoffs at each relevant date and state.

Combine the claims.
Add levered equity and debt payoffs. Confirm that their sum matches unlevered equity, not merely in one selected scenario but in every modeled state.

Compare current prices.
Compare $V_U$ with $E_L+D$. The lower-priced side is bought; the higher-priced side is sold.

Verify the trade.
Show the initial cash receipt and demonstrate that future cash flows cancel state by state.

Worked Example: Arbitrage from Inconsistent Prices

Two firms have identical operating assets and future cash flows. The unlevered firm’s equity sells for $520. The levered firm’s equity sells for $315, and its debt sells for $180. The debt is paid in full in every modeled state, and the levered equity and debt together have the same future cash flow as the unlevered equity. Identify and calculate the arbitrage. Method: compare the prices of equivalent claims, then buy the cheaper combination and short the dearer claim. This avoids assuming that the larger price automatically identifies the portfolio to buy.

Step 1: Price the equivalent claims. Levered equity plus debt costs $315+180=\$495$. Unlevered equity costs $520. Thus the unlevered claim is $25 more expensive.

Step 2: Take offsetting positions. Short the unlevered equity, receiving $520, and buy levered equity and debt for $495. The initial net receipt is $520-495=\$25$.

Step 3: Check future cash flows. If the operating assets pay $Z$, the short position requires payment of $Z$. The long levered equity and debt together receive $(Z-\text{debt payoff})+\text{debt payoff}=Z$. The net future cash flow is $Z-Z=0$ in every modeled state.

Check. The trade produces $25 initially and has zero future net cash flows. If the observed prices persist and the assumptions hold, this is an arbitrage. The conclusion is about the model’s pricing relationship, not a claim that real-world trades are costless or riskless.

Common Traps and Limits

The most frequent calculation error is to match the percentage of equity but not the corresponding debt position. A fraction $\alpha$ of levered equity must be compared with fraction $\alpha$ of unlevered equity, and the associated personal borrowing or lending must also be scaled by $\alpha$. Mixing a full debt amount with a fractional equity holding changes the cash flows.

Common mistake: If an investor buys 15% of an unlevered firm and borrows the full $40 of a comparable firm’s debt, the initial cost is incorrectly calculated as $0.15V_U-40$. The matching construction borrows 15% of the debt value, so the cost is $0.15V_U-0.15D$. The future repayment must likewise be scaled by 15%.

A second trap is to claim an arbitrage after comparing only one scenario. Replication requires equal cash flows in every relevant state and at every date. If debt can default, if personal and corporate borrowing have different terms, or if taxes or transaction costs apply, the simple equality may not hold. The no-tax perfect-market result is a benchmark that isolates the effect of financing; it is not a guarantee that actual investors can borrow on identical terms or execute the trade without constraints.

Check Your Understanding

For each problem, use matching cash flows and prices; assume debt is paid in full in every modeled state unless stated otherwise.

An unlevered company is worth $360, and comparable debt is worth $90 with a one-year payoff of $99. An investor wants 8% of the corresponding levered equity. How much personal borrowing is required, what is the net initial cost, and what is the future payoff if operating cash flow is $Z$? (Answer: Borrow $7.20; net cost is $21.60; payoff is $0.08Z-\$7.92$.)
A levered company has equity worth $260 and debt worth $125. An unlevered counterpart is worth $402. State the arbitrage positions and the initial gain. (Answer: Short unlevered equity for $402; buy levered equity and debt for $385; initial gain is $17, with future cash flows cancelling.)
A levered firm’s equity is worth $438 and its debt is worth $330; identical unlevered equity is worth $750. Which side is overpriced, and what positions produce the arbitrage? (Answer: The combined levered claims cost $768 and are overpriced by $18; short levered equity and debt, buy unlevered equity, and receive $18 initially.)
A levered firm has equity worth $300 and debt worth $100. Its debt pays $110 in one year. An investor wants 20% of unlevered-equity cash flows. Find the portfolio cost and verify its payoff if operating cash flow is $500. (Answer: Buy 20% of levered equity for $60 and lend $20; total cost is $80; future payoff is $0.20(500-110)+0.20(110)=\$100$, equal to 20% of unlevered cash flow.)
An unlevered claim costs $280, while levered equity costs $210 and debt costs $75. The debt payoff is fully met in all states. Give the arbitrage trade and state why its future payoff is zero. (Answer: Short unlevered equity and buy levered equity plus debt; receive $280-(210+75)=\$-5$ by that direction, so reverse the positions: buy unlevered equity and short both levered claims for a $5 initial receipt; the future cash flows cancel because the two sides each pay the same operating-asset cash flow.)

What You'll Learn