# Relevering Beta

When assessing the value of a company’s operations free cash flows need to be discounted using the weighted average cost of capital (WACC). WACC or weighted average cost of capital is calculated using the cost of equity and cost of debt weighting them by their respective proportions within the optimal or target capital structure of the company, i.e.

WACC = E/(D+E)*Cost of Equity + D/(D+E) * Cost of Debt, where E is the market value of equity, D is the market value of Debt.

The target or optimal capital structure of the company is one where WACC is minimized which would result in the maximization of shareholders’ wealth.

The cost of debt can be observed from bond market yields. However the cost of equity may be estimated using the Capital Asset Pricing Model (CAPM) formula, specifically

Cost of Equity = Risk free Rate + Beta * Market Risk Premium

Beta in the formula above is equity or levered beta which reflects the capital structure of the company. The levered beta has two components of risk, business risk and financial risk.

Business risk represents the uncertainty in the projection of the company’s cash flows which leads to uncertainty in its operating profit and subsequently uncertainty in its capital investment requirements.

Financial risk represents the additional risk placed on the common shareholders as a result of the company’s decision to use debt, i.e. financial leverage. This is because with the addition of more debt to the structure the residual claim of the shareholders becomes less certain and hence more risky.

If the capital structure comprised of 100% equity then beta would only reflect business risk. This beta would be unlevered as there is no debt in the capital structure. It may also be known as the asset beta.

To obtain the equity beta of a particular company, we start of first with the portfolio of assets of that company or alternatively a sample of publicly traded firms with a similar systematic risk. We will first derive the betas of these individual assets or firms from market prices. The derived betas are levered betas as they would reflect the capital structure of the respective firms. They would need to be unlevered so as to only reflect their business risk components.

From the unlevered betas a weighted average unlevered beta will be obtained using as weights the proportions of the assets in the company’s asset portfolio or an average across all comparable firms will be derived. The weighted unlevered beta thus obtained would now be re-levered based on the capital structure of the company in order to determine the equity or levered beta for the company, a beta that reflects not only the business risk but also the financial risk of the company.

Un-levering and re-levering beta may be done in a number of ways. A method employed by practitioners gives the relationship between un-levered and re-levered beta as follows:

Levered Beta = Unlevered Beta * (1+D/E), where D/E = Debt-to-Equity Ratio of the company.

The practitioner’s method makes an assumption that the corporate debt is risk free. If corporate debt is considered risky then another possible formulation is:

Levered Beta = Asset Beta + (Asset Beta – Debt Beta) * (D/E) where Debt Beta is estimated from the risk free rate, bond yields and market risk premium.

The above formulations do not incorporate the impact of corporate taxation, i.e. the fact that debt returns tend to be tax deductible. In order to consider the impact of taxation the following adjustments will be made in the relationships given above:

Under the practitioner’s method:

Levered Beta = Unlevered Beta * (1+D*(1-T)/E) where T is the tax rate.

Under the risky-debt formulation:

Levered Beta = Asset Beta + (Asset Beta – Debt Beta) * (D/E)*(1-T).

And WACC would be equal to E/(D+E)*Cost of Equity + D*(1-T)/(D+E) * Cost of Debt.

### Relevering Beta Example

NewCorp is a corporation of personal hygiene and medical subsidiaries. We are estimating its levered beta for the purpose of determining its cost of equity. The personal hygiene subsidiary is worth USD 20 million while the medical subsidiary is worth USD 30 million. The firm has a debt-to-equity ratio of 1. The tax rate for all firms is assumed to be 30%. The risk free rate is 7% and the market risk premium is 6%. The following information has been obtained of firms with comparable systematic risk:

 Comparable Firms Average Beta Average D/E Ratio Personal Hygiene 0.9 20% Medical 1.2 60%

Note that the average betas above denote the average of the levered or equity betas of these firms.

In the first step we will calculated the unlevered betas for each group of firms using the practitioner’s method:

Unlevered Beta for the personal hygiene business = 0.9 / (1+ 0.2*(1-0.3)) = 0.79

Unlevered Beta for the medical business = 1.2 / (1+ 0.6*(1-0.3)) = 0.85

We will then calculate the unlevered beta for NewCorp. This will be the weighted average of unlevered betas where the weights are taken in proportion to the subsidiaries value in the firm, i.e.

Unlevered Beta for NewCorp = 0.79*20m/50m+0.85*30m/50m = 0.82

Levered Beta for NewCorp = 0.82*(1+1*(1-0.3)) = 1.40

Cost of Equity = 7%+1.40*6% = 15.39%.

### WACC, Beta and Market Risk Premium – Industry specific

We can extend the same model to calculate industry specific WACC estimates.

For instance shown below is the calculation for US regional banks, the Computer Services industry and Energy and Power sector using the January 2016 data set shared by NYU and Dr. Damodran on their site.

We have used the default 6% estimate (2015) for Market Risk Premium for all three industries.

A quick sensitivity test of WACC by changing values of Beta and Market Risk Premium shows a range of WACC values between 1.96% – 8.57% for regional banks in the US. This range can become a third input in our in class valuation intuition exercise

While we have used the same risk premium, different Beta and Leverage ratios lead to a different value of WACC for the Computer Services industry and the Energy and Power segment.

As expected a different WACC and a different relevant range because of change in WACC parameters.

WACC calculations for Energy and Power sector.

Sensitivity analysis for Energy and Power sector.

We can now use these estimates as part of our valuation exercises and case studies referenced in Where do valuation multiples come from.  Also see Calculating Beta with respect to Market Indices.

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