## What Is the Irrelevance Proposition Theorem?

The irrelevance proposition theorem is a theory of corporate capital structure that posits financial leverage does not affect the value of a company if income tax and distress costs are not present in the business environment. The irrelevance proposition theorem was developed by Merton Miller and Franco Modigliani, and was a premise to their Nobel Prize-winning work, “The Cost of Capital, Corporation Finance, and Theory of Investment.” It is not uncommon to see the expression adapted to the "capital structure irrelevance principle" or "capital structure irrelevance theory," in the popular press.

### Key Takeaways

- The irrelevance proposition theorem states that financial leverage does not affect a company's value if it does not have to encounter income tax and distress costs.
- The irrelevance proposition theorem was developed by Merton Miller and Franco Modigliani, and was a premise to their Nobel Prize-winning work, “The Cost of Capital, Corporation Finance, and Theory of Investment.”
- The theorem is often criticized because it does not consider factors present in reality, such as income tax and distress costs.
- The theorem also does not consider other variables, such as profits and assets, which influence a firm's valuation.

## Understanding the Irrelevance Proposition Theorem

In developing their theory, Miller and Modigliani first assumed that firms have two primary ways of obtaining funding: equity and debt. While each type of funding has its own benefits and drawbacks, the ultimate outcome is a firm dividing up its cash flows to investors, regardless of the funding source chosen. If all investors have access to the same financial markets, then investors can buy into or sell out of a firm’s cash flows at any point.

This means that in the absence of taxes, bankruptcy costs, agency costs, and asymmetric information, and in an efficient market, the value of a firm is unaffected by how that firm is financed.

Miller and Modigliani used the irrelevance proposition theorem as a starting point in their trade-off theory, which describes the idea that a company chooses how much debt finance and how much equity finance to use by balancing the costs (bankruptcy) and benefits (growth).

Criticisms of the irrelevance proposition theorem focus on the lack of realism in removing the effects of income tax and distress costs from a firm’s capital structure. Because many factors influence a firm’s value, including profits, assets, and market opportunities, testing the theorem becomes difficult. For economists, the theory instead outlines the importance of financing decisions more than providing a description of how financing operations work.

## Example of the Irrelevance Proposition Theorem

Suppose company ABC is valued at $200,000. All of this valuation is derived from the assets of an equivalent amount that it holds. According to the irrelevance proposition theorem, the valuation of the company will remain the same regardless of its capital structure i.e., the net amount of cash or debt or equity that it holds in its account books. The role of interest rates and taxes, external factors that could significantly affect its operational expenses and valuation, in its account book is completely eliminated.

As an example, consider that the company holds $100,000 in debt and $100,000 in cash. The interest rates associated with debt servicing or cash holdings are considered to be zero, according to the irrelevance proposition theorem. Now suppose that the company makes an equity offering of $120,000 in shares and its remaining assets, worth $80,000, are held in debt. After some time, ABC decides to offer more shares, worth $30,000 in equity, and reduce its debt holdings to $50,000.

This move changes its capital structure and, in the real world, would become cause to reassess its valuation. But the irrelevance proposition theorem states that the overall valuation of ABC will still remain the same because we have eliminated the possibility of external factors affecting its capital structure.