P = D / [r - g]

Where:- P - Current price of equity
- D - Dividends expected in next period
- r - Required rate of return on the market
- g - Divided growth rate

Then divide both sides by earnings per share, E:

P/E = (D/E) / [r - g]

Note that we now have an equation for a PE ratio (with dividend over earnings per share as the dividend payout ratio - a relationship I hinted at in my previous post).

P/E = (Dividend payout ratio) / [r - g]

Now in my mind, I wanted to see if I could simplify the formula even more. I knew there was a relationship between dividend payout and growth (as I mentioned in my post about RIM issuing dividends, a growing company will have a higher retention ratio and keep more cash on hand).

Sustainable Growth

Recall that the earnings retention ratio is one minus the dividend payout ratio (and vice versa). And that the sustainable growth rate is equal to the return on common equity x the earnings retention ratio.

Dividend Payout Ratio = 1 - Retention Ratio

Sustainable growth rate = Return on Common Equity x Retention Ratio

Retention Ratio = Sustainable growth rate / Return on Common Equity

Sustainable growth rate = Return on Common Equity x Retention Ratio

Retention Ratio = Sustainable growth rate / Return on Common Equity

Now, assuming a constant dividend payout ratio, with constant dividend growth, a logical assumption is that the sustainable growth rate is also the same (the elements in the model all grow at the same rate, retention, dividend etc). Also, we assume that the return on common equity is the same as the required rate of return on the market. So what do we get?

Dividend Payout Ratio = 1 - (Sustainable growth rate / Return on Common Equity)

Assuming:

- Sustainable growth rate, g
- Return on Common Equity, r

Dividend Payout Ratio = 1 - (g/r)

Substituting back into our formula for PE we get:

P/E= [1-(g/r)] / [r-g]

PEG

I'm excited about this formula because it uses only two factors to determine the price to earnings multiplier and the two factors are expressed entirely in rates of growth. If you simplifying the formula by expressing the numerator with a common denominator (r), you get:

P/E = [(r-g)/r] / (r-g) or

(r-g) / r (r-g)

(r-g) / r (r-g)

Notice that the (r-g) terms cancel and you are left with:

P/E = 1 /r

Look familiar? It should. This is the mathematical justification for PEG ratios where r is the Return on Equity (equity growth). It also justifies my previous post about creating your own virtual dividend policy (because notice that the dividend policy no longer affects the price of the equity using PE ratios).

Payback Period Model

Another interesting point is when looking at P/E through the glass of payback period. In essence: considering my initial investment (the price of a share), how long will it take to make enough earnings to pay back my initial investment (excluding discounting)? You can see the symmetry in the formula as price is the numerator in both formulas, expressed as a dollar price on the left and a percentage (100%) on the right. The earnings is the denominator in both, expressed as a dollar value on the left and a percentage (expected annual return in the form of growth) on the right.

In other words, each year, you gain r percent back. Your payback period will be 100% / r.

CAPM

Also note that using CAPM, r can be determined through the SML by:

r = E(R) = RFR + beta [Rm - RFR]

Where:- E(R) is the expected rate of return (which relates to the cost of equity)
- RFR is the rate of return for a 'risk free asset' (approximated with treasury bill rate)
- Rm is the return for the market portfolio (or in this case, security)
- beta is the standardized measure of systematic risk.

I consider this to be the first principles of financial valuation.

## 4 comments:

There is an error in your reasoning. The r that you began your argument is the r for cost of equity and not ROE. since those r's are not identical, you cannot cancel them out. 1/r (r is the cost of capital) would be the P/E ratio for a zero growth firm.

That is a great point. However, even if they aren't identical, I believe they are fairly similar to the point where they can be used as proxies for one another. This was the case in the M&A question that was asked during our stock pitch competition more recently:

http://amgstr.blogspot.com/2009/11/using-proxies-m-accretion-or-dilution.html

The reason this works is because the inverse of P/E would be E/P or a proxy for r. Essentially, it's asking how much earnings (return) will you get per dollar invested (similar concept to cost of equity).

I agree with you that it can be a proxy but it is only a lower bound because the E/P works if you assume zero growth rate.

Would you use 19% (ROE) as your cost of equity for a company like PG? What about .06 which is close to the E/P value today?

Thanks for sharing this useful info. Keep updating same way.

Regards,AshishRelationship Building-

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