Levering and Unlevering Beta - Mechanics of the Model

It occurs to me that I didn't really describe the formula for levering and unlevering Beta and why it works. Let's have a closer look:

First think of any given industry. There are some major assumptions required to get levering and unlevering beta to work. For instance, a dollar invested in an industry will give a constant return proportional to it's risk (beta). This is one of the fundamental assumptions in CAPM.

Levering beta is the idea that I can amplify the result of a deal by using debt. Let's look at an example:

I'm in an industry that has a beta of 1.2

ERP is 5%

RFR is 4%

ke = RFR + beta * ERP = 4% + (1.2) 5%

= 10%

So if I invest a dollar, I can expect to get a return of 10%.

However, I can use leverage (borrow money) so that I want to experience an amplified result:

I'll invest 1 dollar, but allow borrow another dollar to invest with. So I have two dollars in play, but only one of those is mine. My return? Well, assuming you can borrow at RFR (a huge assumption), you'll gain another 10% on that additional dollar, but have to pay 4% (RFR) so you'll gain another 6% on top of your original 10% for a total of 16%. You've doubled your systematic risk by having two dollars in play. Let's check with CAPM:

Beta = 2*1.2 (twice as much exposure)

ke = RFR + beta * ERP

= 4% + 2.4 * (5%) = 16%

This makes sense. For every additional dollar you borrow at RFR and invest in the industry, you will make the spread between their returns (or the beta * ERP).

So if you borrow D, dollars (D in this case stands for Debt), versus E dollars of your own money (Equity), and because CAPM is a linear function, you can expect:

• Your exposure will be based off of D+E dollars (total capital in the game - systematic risk, exposure)
• But off a base of E dollars invested (the money you have invested yourself, or Equity)

So if an industry has a beta of Ba, and you use leverage D on your initial investment E, you can expect your new beta of Be, to be:

Be = Ba * (E+D)/E

Note, however, that (E+D)/E = 1 + (D/E)

Look familiar? It's the levering formula:

Be = Ba * (1 + D/E)

But there is still one piece missing. The (1-t) portion of the formula is to simply account for the fact that debt that you borrow provides a tax shield, so the final version of the formula replaces D with (1-t)*D to get:

Be = Ba * (1 + (1-t)*D/E)

Hope this decomposition helps. I noticed this relationship is very prevalent in financial calculations as Anita McGahan clarified in one of her lectures on Starbucks in first year when talking about Operating Strategy versus Financial Strategy in the Dupont Decomposition. This was something I had also wondered about previously.