Leverage is quite often used to improve returns. When evaluating a business, it is important to assess underlying performance with as well as without the impact of leverage. To measure performance without the impact of capital structure, we need un-levered Beta or Asset Betas. (*See the difference between Alpha and Beta*).

## Why do we need to relever Beta?

From a valuation perspective in a buyout opportunity there is quite often a dramatic change in capital structure, specially within Leveraged and Management Buy outs (MBOs and LBOs). Within this context the current capital structure is not relevant for the buyer. Beta therefore first needs to be un-levered to get to Asset Beta and then Asset Betas are re-levered again to test the impact of multiple debt levels (proposed capital structure) to find an optimal mix between risk, returns and debt pay down schedules.

When assessing the value of a company’s operations free cash flows are discounted using the weighted average cost of capital (WACC). WACC or weighted average cost of capital is calculated using cost of equity and cost of debt weighting them by 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.

Cost of debt can be observed from bond market yields. Cost of equity is 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.

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

## How do we relever Beta?

To obtain the equity beta of a particular company, we start with 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 un-levered 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

BetaCorp is a corporation that has two primary business lines – personal hygiene and consumer off the shelf pharmaceuticals. 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 consumer pharma 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 consumer pharma business = 1.2 / (1+ 0.6*(1-0.3)) = 0.85

The Beta for BetaCorp will be the weighted average of un-levered betas where the weights are taken in proportion to the subsidiaries value in the firm, i.e.

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

Levered Beta for BetaCorp = 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 in * Where do valuation multiples come from*.

Outside of the valuation and WACC world if you are interested in using Beta and Alpha in portfolio construction, portfolio allocation and portfolio optimization models, s*ee Calculating Beta with respect to Market Indices and Calculating Alpha and Beta*