Sorry Mike - I was really rushed as you can tell.
Here it is again without skipping any steps.
OK first some assumptions. The first key assumption is that a company's earnings (E) are not a pro-forma fiction of financial engineering, and are representative of the business model.
The second is that the company will grow proportionately at some average rate (G) so that all key ratios will be preserved. Gross margins, net margins, inventory, accounts receivable and so on.
Next we make some economic assumptions.
First, that the price of the company should be some multiple (k) of discounted cash flows
P = k * DCF
Second, that we will discount at some rate r (which can be the bond rate or wacc or whatever). So if CF(n) is the cash flow in year n we have:
P = k * SUM [CF(n) * 1/(1+r)^n]
Now, our economic ratios are preserved, remember that? So CF(n)/Sales(n) is constant. So CF(n) = c * S(n) where c is the ratio of cash-flow to Sales in year n... S(n). Our equation becomes
P = k * SUM [c * S(n) * 1/(1+r)^n] or
P = k * c * SUM [S(n) * 1/(1+r)^n])
Next we know that the company is growing at rate G, so S(n) = S(n-1) * (1+G). Or S(n) = S(0) * (1+G)^n, where S(0) is the Sales in year 0.
Which gives us
P = k * c * SUM [(S(0)* (1+G)^n)/((1+R)^n)] or
P = k * c * S(0) * SUM [((1+G)/(1+R))^n]
One final substitution is in order, then some implications.
Recall that the financial ratios like net margin are constant. So Earnings (E) grow with sales: E = C * S where C is this constant ratio.
Which we can substitute
P = k * c * C * E(0) * SUM[(1+G)/(1+R))^n]
Now, k is a constant. So is c and C and so is that icky sum term (once we have chosen G and R and n).
Which leads to P = K * E(0) <K is a constant equal to all those other constants multiplied together>
So you can favorably compare PE ratios [EDIT: this constant K we just derived is the PE Ratio] between companies that have similar values of k, c, C, G and R. One should expect different PE ratios if these numbers are different.
Hopefully, this shows simultaneously that (a) PE has a mathematical basis in economic reality and (b) how many assumptions are going unsaid when someone says "XYZ only has a PE ratio of 23", and (c) how careful one must be comparing PE ratios across companies.
Next, let's look at PEG.
We have P = E * Z * SUM[[(1+G)/(1+R))^n] where Z = k* c * C * S(0)
You see that term "G" is already embedded in the equation. But not as some linear co-efficient. Sadly, it is embedded within a power series with two other variables. We could waste a lot of time trying to tease it out. I leave it to the thread to see if you can, but even in the degenerate case where R=0 there is both a power term and a linear term.
Thus I think it is pretty clear my assertion that PEG is an economic fiction, while PE at least is a cousin of DCF (several assumptions removed).
Happy father's day.
John |
