To understand how a company's financing choices affect its value, we first need to imagine a perfect world. This isn't the real world, but a simplified model where the core mechanics are easier to see. This theoretical space is called a perfect capital market.

In this market:

• There are no taxes.
• There are no bankruptcy costs. If a company fails, assets can be sold for their true market value without any legal or administrative fees.
• There are no transaction costs for buying or selling securities.
• Everyone has the same information about a company's future prospects.
• Individuals and companies can borrow money at the same interest rate.

By stripping away these real-world complexities, we can isolate the effects of capital structure itself. This framework was developed by Franco Modigliani and Merton Miller, and their insights form the bedrock of modern corporate finance.

Modigliani and Miller's first proposition is simple but profound: in a perfect market, a firm's value is determined by the cash flows generated by its assets, not by the mix of debt and equity used to finance them.

The value of a firm is the present value of its future earnings. The capital structure simply carves up how those earnings are distributed between debt holders and equity holders. Think of the firm's total value as a pizza. Proposition I says the size of the pizza doesn't change whether you slice it into four, six, or eight pieces. The total amount of pizza remains the same.

This is often called the 'law of conservation of value'. The total value of the assets must equal the total value of the claims against those assets (debt and equity). Let's say we have two identical companies: Firm U is unlevered (100% equity) and Firm L is levered (a mix of debt and equity). M&M Proposition I states that $V_L = V_U$, where V is the total market value of the firm.

What keeps their values equal? An arbitrage process called home-made leverage. If Firm L were somehow more valuable than Firm U, an investor could sell their shares in Firm L, buy shares in Firm U, and borrow money on their own to replicate the exact same pattern of cash flows and risk they had with Firm L, but with some cash left over. Since all investors would do this, the prices would adjust until the firms had the same total value.

If debt is usually cheaper than equity, why doesn't adding more debt make the firm's overall cost of capital cheaper? This is where M&M's second proposition comes in. It states that the cost of a firm's equity increases in direct proportion to its debt-to-equity ratio.

The reason is risk. Debt holders have a priority claim on the company's earnings and assets. They must be paid their interest before shareholders get anything. As a company takes on more debt, the risk for shareholders goes up. They are last in line to get paid, and the slice of earnings available to them becomes more volatile. To compensate for this higher risk, shareholders demand a higher expected return. This increased cost of equity perfectly offsets the benefit of using cheaper debt.

This relationship is captured in a precise formula. First, let's distinguish between two types of risk:

• Business Risk: The inherent risk in the firm's operations, determined by its assets and the industry it operates in. This risk exists even with no debt. It's represented by the firm's unlevered cost of capital.
• Financial Risk: The additional risk placed on shareholders as a result of using debt. It's the risk of not being able to meet debt obligations.

The cost of equity is the combination of the return required for business risk and the extra return required for financial risk.

As a firm adds more debt, the debt-to-equity ratio increases, which drives up the cost of equity. This linear increase in the cost of equity exactly cancels out the benefits of using more of the cheaper debt. The result? In a perfect market, the firm's Weighted Average Cost of Capital (WACC) remains constant, regardless of its capital structure. It is always equal to the unlevered cost of capital, $r_0$.

The M&M propositions provide a crucial baseline. They show that in a simplified world, financing decisions are a sideshow. The real driver of value is the quality of a company's assets and its operating decisions. In later sections, we will relax the perfect market assumptions and see how things like taxes and financial distress costs change this fundamental picture.