CFA CFA Level 2 Deriving justified P/E expressions

# Deriving justified P/E expressions

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• I must be doing something algebraically wrong here but I can’t figure out how the justified P/E is derived.
Could someone explain how we get to the following expressions:

The CFA institute lists both expressions as:
Leading P0/E1 = (D1/E1)/(r-g)
Trailing P0/E0 = [D0(1+g)/E0]/(r-g)

Thanks

• 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).

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