@clangerh
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Assume g is constant the entire time, per the Gordon Growth Model.
P0 = D1/(r-g) and keeping proportional, P0/E1 = D1/(r–g)/E1 or think of it as P0/E1 = D1/(r-g) * 1/E1 and there’s your first equation.
Now, since D1/E1 is the payout ratio, b, or the complement of the retention ratio, (1 – rr), you have your justified leading P/E1, where P0/E1 = (1 – rr)/(r-g).
Again, keeping proportional, P0/E1 = P0/E0*(1+g) so P0/E0 = (1 – rr)/(r-g) * (1+g) which you normally see cleanly as (1-rr)*(1+g)/(r-g). And based on D0/E0 being the percent of what’s paid, b, it’s again (1 – rr). So, swap the (1 – rr) for D0/E0 on that last equation and get D0(1+g)/E0/(r-g).
Think about how this concept forces you to look at a company as if every single one out there has two betas. One shows up on Yahoo! Finance, and the other is only known to a company’s management. These betas are just ways of describing the stock. Since in finance the easiest way to express risk is with volatility (usually with standard deviation), remember how beta represents this relationship.
The unlevered beta represents the core business that the company is involved in. The levered beta represents this core business but now layers on the fact that the company is taking on debt. If there was risk (i.e., beta) before, now it’s going to be even higher. The levered beta describes how the company is taking on risk in two forms: operational risk (running the plant, say) and financial risk (missing/canceling bond payments).
If I want to scrutinize a company that makes widgets (maybe in order to take it over myself or to see how the market views widget makers because perhaps Berkshire Hathaway is going to buy them, and I own shares of BRK) I probably want to unlever the beta to see the core riskiness of making these widgets. But, when I look at that undressed beta of the widget maker, my own company’s beta might seem high. Since if I were to take over the widget makers I would definitely need to take on a little debt to keep things running, the most realistic way to make it a one-to-one comparison is probably to re-lever the widget maker’s core/unlevered beta with my D/E ratio (and possibly tax rate, but don’t worry about including that or not—you don’t include it in Level II).
Now we can see how risky making widgets is, and can do so without the widget maker’s executives financing decisions getting in the way (…at least in theory…)
Here’s a useful resource explaining the relationship between debt and beta: http://www.cs.princeton.edu/~kazad/resources/finance/betaborrowing.htm
