 # Levered and Unlevered Beta - Theoretical Challenge

I have a somewhat theoretical question. I understand that most people aren’t here for that, so feel free to move on if you’re just here to get letters next to your name… When we try to apply a beta to thinly traded or private equity, we have the option to use comparable beta’s, but, we have to make sure that the beta’s represent the target company’s capital structure. So, we unlever and relever the comparable Beta… Ok, so why do we use the (1+D/E) to unlever and relever as opposed to (1+D/(D+E))?

beta debt = 0 so Beta-A = Be * E/(D+E) so Be = (D+E)/E * Beta-Asset = [1+D/E]* Beta-Asset Unlever => Beta-Asset = Beta_Equity/ (1+D/E) New D/E available Beta-ENew = Beta-Asset * (1+D/E New)

Nice! I figured you would be the one to answer this one. So we can just go: (B-Levered-for-comparable)*(proportion of equity in comparable) = B-Unlevered (B-Unlevered)/(proportion of equity in target firm) = New Beta for Target Firm So much better… Thank you!

oh, many thanks CP, I have lined up this area to be fine tuned. Johnny, be careful with the way you use * and /. Ensure you understand when you use * or / with the way you define proportion of Equity. Observe the sign changes depending on whether you use 1+(d/e) or e/(d+e)

oal29, there is no sign change involved 1+d/e = (d+e)/e = 1/(e/(d+e)) watch for the brackets. both are the same sign…

Naturally debt does not have a beta. But theoretically speaking, why should a stock with higher levels of debt react (or follow) to market moves more closely? I guess the earnings will gain some volatility and the already existing relationship to the market will intensify… Am I seeing this straight?

I don’t think you are looking at this correctly; Beta(asset) = B(equity) * E/(D+E) or put another way (0 + E)/(D+E) b/c B(debt)=0 So as E gets lower, the equation (0 + E)/(D+E) approaches 0 which causes Beta(asset) to approach 0 as well.

I understand that. Its not what I meant, but I answered my own question going through an example… In case you’re interested, here it is: A company has 100% equity and a beta of 1.4. It then decides to go 50% debt and 50% equity, it does this by issuing debt and buying back shares. According to the equation, the beta will now be: 2.8. So, the math was obvious, but I was having a difficult time understanding why the stock price would now react with such added vigor (higher beta). After thinking about it for a few minutes, I realized that nothing changes for the debt holders as the market moves up and down (and as the company’s value moves up and down), and that shareholders now get all the reward or punishment going forward. In this example, the reward/punishment doubles as they have to assume twice as much responsability for failures and successes…

I think you guys may be overcomplicating this. while debt may have a beta of 0, it obvioulsy affects an assets beta, otherwise there would be no need to deleverage. to delverage beta you multiply the proportion of contributed capital provided by equity by the unleververed beta: Bu = Bl * E/(D+E) . If you simply divide each term in the equation E/D+E by E you get 1/(D/E+1).

I think you guys may be overcomplicating this. While debt may have a beta of 0, it obvioulsy affects an asset’s beta, otherwise there would be no need to deleverage. to deleverage beta you multiply the proportion of contributed capital provided by equity by the levered beta: Bu = Bl * E/(D+E) . If you simply divide each term in the equation E/D+E by E you get 1/(D/E+1).