Matters of investment and the stock market rely on how both the market and the company being considered are perceived. The cost of equity formula calculates the returns investors would require before putting resources into a company and can be calculated with (levered) or without (unlevered) debt and equity. They are both shaped by the volatility of the market and the market value's ratio of debt to equity.

## What Is the Levered and Unlevered Equity Beta Formula?

This article from the Warsaw School of Economics explains that while it's difficult to estimate a new company's equity or the equity of a company's new project based on its own merits, you can find a company or project as similar as possible to the new one and evaluate how it performed.

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This is described as a "pure-play" company and the known values from this pure-play are used to estimate the new company or project. The Beta (ß) coefficient represents the volatility of the company's stock returns relative to the market's returns. The Beta of unlevered equity, ß_{U}, is calculated thus: ß_{U} = ß_{Equity} / [1 + ( 1 - T_{pure-play})(D_{pure-play} / E_{pure-play})], where D represents the market value of debt, E represents the market value of equity and T is the tax rate as a decimal. As the debt-to-equity ratio increases, so too does the equity risk, which causes the cost of equity to rise.

## Rate of Return on Levered Cost of Equity Formula vs. Unlevered Cost of Equity Formula

A detailed article on ResearchGate by Joseph Tham and Ignacio Valez-Pareja explores the cost of levered equity in terms of the Weighted Average Cost of Capital (WACC) and Capital Asset Pricing Model (CAPM).

In particular, the CAPM Levered return on equity formula goes as follows: R_{E} = R_{F} + ß_{E} (R_{M} - R_{F}), where R_{E} = levered cost of equity capital, R_{F} = risk-free return, ß_{E} = levered beta (volatility of levered stock's return relative to market's returns), R_{M} = expected return on market portfolio and (R_{M} - R_{F}) = market risk premium for bearing one unit of market risk

In turn, the unlevered rate of return equity formula uses much of the same kind of data: R_{U} = R_{F} + ß_{U} (R_{M} - R_{F}), where R_{U} = expected rate of return on stock of unlevered company and ß_{U} = unlevered equity beta (all-equity beta).

## Value of Levered and Unlevered Firms

The value of a levered firm is equal to the value of an unlevered firm plus the present value of tax shields, less the present value of bankruptcy costs, where a tax shield can be calculated with the formula: Tax shield = T * I, where T = income tax rate, I = interest payments = (R_{F}D) and D = market value of long-term debt.

So, an unlevered firm is inherently riskier, as long-term debts can increase the tax shield and having leverage means having more assets with which to realize gains. A less volatile market, represented by a lower Beta value, will increase rates of return on both levered and unlevered companies. Higher Beta values can be offset by leverage and tax shields.

**Consider also:** Difference Between the Full Equity & Partial Equity Method