Comprehensive Guide to Beta in the CAPM and Cost of Capital

I've been wanting to do a post like this for a while given the amount of confusion regarding terminology and the various formulas during IB interview prep. All from Berk&DeMarzo's Corporate Finance and an MBA Corporate Finance class. Note that this is academic and probably overkill, but I hope it's useful to some.

Basic idea:
1. Calculate unlevered asset beta (various methods depending on assumptions)
2. Use this beta and CAPM to calculate unlevered asset cost of capital aka unlevered equity cost of capital
3. Use this cost of capital as the discount rate for the APV method OR calculate levered equity cost of capital using M&M2 to put in the WACC formula

Step 1

Comparable company has no debt in capital structure?

Easiest case. Manually estimate equity beta ("market model" methodology)

• Run regression where the dependent variable is individual stock's return and explanatory variables are a constant and stock market's contemporaneous return.
• Because no debt, this beta is also unlevered asset beta.

Comparable firm has non-zero debt?

This is where things get more complicated. Must estimate equity beta but also the debt beta and debt tax shield beta to get unlevered asset beta.
Because the following is true:
1. value of assets = value of equity + value of debt
2. portfolio of x in asset a and (1-x) in asset b mathematically has portfolio beta = xBa + (1-x)Bb
then

where Ba is levered asset beta.
If we assume debt beta = 0, second term disappears. Otherwise, you can research data tables for average debt betas by rating and maturity. For example, if company's senior unsecured debt is rated A, use the corresponding average beta (about .05). From here calculate levered asset beta. We still need unlevered asset beta.

We also know asset value of levered company = unlevered asset value + debt tax shield value, so:

How we simplify this and solve for Bu depends on the debt policy:

Constant debt-to-value ratio

• Let's say unlevered company value increases by 1%, debt oustanding increased by 1% and debt tax shield increases by 1%
• Therefore tax shield is proportional to unlevered firm, so tax shield beta = unlevered asset beta
• Substituting B taxshield = Bu, above formula simplifies to just Ba = Bu.
  Therefore with constant debt-to-value:
  
  The unlevered asset beta is equivalent to the levered asset beta.

Debt is independent of unlevered value

Unfortunately, the above case is not typical. A common approximation to simplify the original formula:

and solve for Bu is to assume tax shield beta = company debt beta. With algebra, this reduces to:

Moreover, if the company holds on to the a constant dollar value of debt forever, using a perpetuity formula for the PV of the debt tax shield: is DrT/r = DT = V taxshield. Substituting:

Notice that if you set debt beta = 0 and manipulate by multiplying both sides by E + D(1-T) and dividing by E, you get that "unlevering the beta" formula on Investopedia that everyone tells you to memorize:

Step 2

Now that you have unlevered asset beta, you can plug this into the CAPM formula to get unlevered asset cost of capital, which importantly assumes 100% equity financing.

Step 3

With unlevered asset cost of capital, you can use that as the discount rate with APV method or use it to calculate WACC. With the WACC method, use this cost of capital to get the levered cost of equity given the firm's cost of debt and leverage ratio with M&M Proposition 2

Finally, discount unlevered FCF using after-tax WACC