# How to Calculate the Beta of a Combined Firm

Beta is a measure of how much a stock moves relative to movements in the overall stock market. Technically, it is the covariance of returns of the stock and the overall market (represented by an index such as the Standard & Poor's 500) divided by the variance of the market.

## Meaning of Beta

If a stock has a beta of 1.0, then if the market goes up one point, the stock will also go up one point. If a stock has a beta of zero, an upward or downward movement in the market will result in no movement in the stock. If a stock has a beta of negative 1.0, if the market moves up one point, the stock will move down one point. If a stock has a beta of 2.0, if the market moves up one point, the stock will move up two points.

Keep in mind that beta is measured based on historical returns, meaning that beta is not a perfect indicator of future performance. If a stock has a two-year beta of 1.0, what this really says is that over the past two years, when the market has moved up one point, the stock also has moved up one point.

## Weighted-Average Beta

If two firms are merged into one firm, the combined firm's beta is based on the weighted average of the market capitalizations of the two predecessor firms. Market capitalization refers to the total equity value of a company and is calculated by multiplying the number of shares a company has outstanding by each share's market value, or trading price. Alternatively, you can estimate a company's market value by performing a valuation of its equity, usually by applying a valuation multiple to a metric like earnings or revenues. For example, the price-to-earnings ratio is one of the better known valuation ratios. If a company records earnings of \$1 million and the average price-to-earnings ratio of comparable companies is 10.0, the equity value of the company would equal \$10 million (P/E ratio of 10.0 multiplied by annual earnings of \$1 million).

If two firms combined via a merger and each company had a market capitalization of \$1 million, the total market capitalization of the newly combined firm equals \$2 million. If Firm A has a beta of 1.0 and Firm B has a beta of 2.0, the newly combined firm's beta equals 1.5 (1/(1+1) multiplied by 1.0 plus 1/(1+1) multiplied by 2.0).

Using the same predecessor firm betas, if Firm A's market capitalization equaled 25 percent of the newly combined entity's market capitalization, Firm B's market capitalization must equal 75 percent. In this case, the newly combined firm's beta would equal 1.75 (0.25 multiplied by 1.0 plus 0.75 multiplied by 2.0; or 0.25 plus 1.5).

## Using Unlevered Betas

The above example does not account for different debt levels and tax rates affecting the predecessor firms. If two companies are held in the same portfolio as passive investments, simply calculating the weighted-average beta is sufficient. However, if the two firms are combined so that a new tax rate and capital structure (the mix of stock and bond financing used by the company to raise cash) is applicable, the new firm beta is calculated using unlevered betas. Unlevered beta is calculated by dividing the observed beta by: [1+(1 minus the tax rate) multiplied by (debt/equity)]. For example, using a tax rate of 35 percent, debt of \$5 million and equity of \$10 million, unlevering a beta of 1.0 requires the following calculation: 1.0 divided by [1+(1-0.35) multiplied by (5 million/10 million)], or 1.0/(1 + (0.65 multiplied by 50 percent), which equals 0.75. Remember to multiply the (1 minus the tax rate) by (debt/equity), and then add the result to 1 in the denominator.

Debt to equity, the second part of the equation, is expressed in dollar terms, and debt can be pulled directly from the balance sheet. Beta can be obtained from a wide variety of sources, many of which are online and free. Once you have unlevered each predecessor firm's beta, you can calculate the weighted-average based on market value of equity.

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