As we have seen, no matter how many years of a company’s earnings we valiantly endeavour to forecast, most of our valuation resides in the terminal value. After all, what are 5, 10 or even 20 years for a company that is assumed to live forever?

Imagine then that, wisened up to this fact, we give up on our N-period forecasting effort, set N=0 and therefore:

In this simplified model, where, right from the start, dividends grow forever at a constant rate G, the dividend yield D_{t+1}/P_{t} is also a constant, equal to R-G. Assuming a constant retention ratio H=1-D/E, earnings E also grow at G, and the forward price/earnings ratio P_{t}/E_{t+1} is itself a constant, equal to (1-H)/(R-G). As long as the company distributes some dividends (H<1), the PE ratio is positive, but it can be any number, depending on the gap between R and G.

This is not very useful. However, there is a relationship between earnings growth and profitability. The company’s book value B_{t} is increased by retained earnings and new equity issues. Assuming the latter to be zero in steady state, we have: B_{t+1}=B_{t}+E_{t+1}-D_{t+1}=B_{t}+H∙E_{t+1}. Therefore, a constant return on equity ROE=E_{t+1}/B_{t} requires B_{t+1}=(1+H∙ROE)B_{t} and therefore G=H∙ROE: earnings growth is a linear function of profitability, with slope H. The retention ratio measures the trade-off between growth and dividends. If H=1, earnings are always entirely retained and reinvested in the business, G=ROE but PE=0: the company is worthless to shareholders. If H=0, earnings are always entirely distributed to shareholders, but the company does not grow: G=0 and PE=1/R.

Taking this relationship into account, the PE ratio can be rewritten as:

PE is a linear function of GO=G/(R-G), with slope FF=(ROE-R)/(R∙ROE) and intercept 1/R.

GO represents Growth Opportunities: it is the present value of equity increases accruing to the company from future investments, as a percentage of current equity:

FF is the Franchise Factor: it measures the PE impact of future investment projects earning a return on equity above the cost of capital.

1/R is the base PE: it is the PE ratio of a growthless company (G=0) where earnings are entirely distributed to shareholders (H=0).

This is a more useful relationship. It shows the PE ratio as the sum of a base PE, equal to the inverse of the cost of capital, and a premium PE, reflecting profitable Growth Opportunities. Likewise, the company’s valuation can be seen as the sum of a Tangible Value TV=E/R and a Franchise Value FV=FF∙GO∙E. More on this decomposition can be found here.

The Franchise Value, and with it the premium PE, owe their existence to the spread between ROE and R. Only a positive franchise spread can justify a PE above base. Without it, the company can still grow earnings and reinvest them in the business, but its valuation will not differ from the valuation of a growthless company that distributes all its earnings to shareholders. In that case, since ROE=E/B and TV=E/R, ROE=R implies TV=B.

In a one-period model (N=0), the franchise spread is assumed to last forever. But this is unlikely to be the case: in reality, investment opportunities are finite and ROE decreases over time, eventually driving the Franchise Value to zero. This is why we need N>0: from 0 to N, the company uses its franchise, invests at ROE>R and grows at G*=H*∙ROE, where, in contrast with the one-period model, G* can be higher than R and can even approximate ROE as, in order to maximise growth opportunities, the company reinvests most of its earnings. After N, however, the company will exhaust its franchise and continue to invest at ROE=R, growing at G=H∙R.

Let’s leave FV aside then, and focus on TV. The Tangible Value is the ratio of forward earnings to the cost of capital, and does not depend on the franchise spread. To obtain a meaningful figure, rather than just taking forward earnings at face value we should endeavour to estimate *normalised* earnings – earnings cleared of transient components and reflecting the company’s earning capacity in normal market conditions. This is easier said than done, but that’s where analysts can truly prove their mettle. Once we have a reliable measure of normalised earnings, TV still depends on R. But, as shown in the following figure, TV is much less sensitive to R compared to FV:

The figure is drawn with E=4, ROE=12% and H=50%, hence G=6%. If R=8%, we have TV=50 and FV=50: the company is worth 100, the base PE is 12.5 and a Franchise Factor of 4.2 transforms Growth Opportunities of 3 into a PE premium of 12.5, for a total PE of 25.

Of course, by changing the model’s parameters we can still get any number we want. Move R to 9%, for example, and the valuation drops by a third; move it to 10% and it halves. But notice that most of the drop comes through the Franchise Value: raising R from 8% to 10% moves FV from 50 to 10, but TV only moves from 50 to 40. Raise R further to 12% and FV goes to zero (as R=ROE), while TV drops to 33. Once we get past 12% FV turns negative, as the company is investing in projects that earn a return below the cost of capital. In that case, the valuation falls below TV, which is itself lower.

What is the true cost of capital? While we are not short of often fanciful ways to come up with a number, we don’t really know. But what we do know is that, as long as the company is not embarking in value-destroying investment projects, the Tangible Value sets a floor to the company’s valuation: the company is worth at least TV.

Imagine then that the market is pricing the company at 50. Our one-period model tells us that the company is worth 100. This sets the Franchise Value at 50, which may be too high, as the franchise spread is unlikely to last forever. If our normalised earnings estimate is right, however, we know that, at a price of 50, the market is placing no value at all on the franchise: 50 is the value of a growthless company that distributes all its earnings to shareholders or, equivalently, a company with no franchise, fruitlessly investing at the cost of capital.

To put it the other way around, at 50 the market is placing a multiple of 12.5 on our normalised earnings estimate, implicitly assuming a cost of capital of 8%. The cost of capital may go up, as the market reassesses the risks associated to the company’s cash flows. But, as the figure shows, a higher R has a limited impact on TV. Of course, there will always be a cost of capital large enough to drive the valuation all the way to zero. But as long as we can trust that R will not exceed the company’s ROE, we can expect a limited downside, significantly lower than the upside deriving from the unrecognised franchise. In this sense, the Franchise Value is none other than Graham’s Margin of Safety.

A low multiple on normalised earnings is a strong message that even a simple one-period model can deliver.

Hi Ted – I didn’t say that the company is fruitless, but that its investment is fruitless, in the sense that it merely earns the cost of capital. The company grows at G=HR, but it is worth as much as growthless company (G=0) that distributes all earnings to shareholders (H=0). For example: 2/(8%-4%)=4/(8%-0%)=50. Only a franchise spread (ROE>R) can create a Franchise Value, i.e. P>B.

A neat little article, Max.

Although I can’t see how a company investing at the cost of capital is fruitless. If RoE=7%, and R=7%, (and thus P=B) the investors receive a return of 7% from their capital being invested at 7%. Capital is thus fairly rewarded. So too (we assume) are labour and resources, pre-profit. Fruit for all! Perhaps fair pricing is fruitless for cunning value investors seeking excess profits, which i guess is your point…

Ultimately the market wants an interpretation of P/E that is simple to apply and calculate. I think franchise adjustments are mathematically accurate, but still too complex for market adoption. The most simple technique is to calculate P/Es using visible E forecasts from the next three years and then assuming 100% future payouts. This undervalues persistent high growth stocks, but the effect is marginal for almost all stocks. For high growth stocks an assumption of discount and growth convergence is a simple fix. That’s my two cents worth, have a good easter!