# Greenwald Formula

I’m nearly through Bruce Greenwald’s “Value Investing” book.

The valuation framework discussed in the book (net asset value, earnings power value, etc.) is both philosophically and quantitatively different from the DCF calculations that I’ve been practicing off of Aswath Damodaran‘s book and notes.

In his book, Greenwald lays out a simple relation between value, profitability, and growth. This is an extremely useful check on intuition.

## Earnings Power Value

Earning power value is simply the adjusted after-tax earnings (E) divided by the cost of capital (COC). $EPV = \dfrac{E}{COC}$

The earnings are typically adjusted for maintenance capex, non-recurring expenses etc. EPV is a no-growth estimate of the value of the firm. For an example, consider this recent valuation of National Oilwell Varco.

The return on capital (ROC) is defined as the ratio of the adjusted after-tax earnings (E) and the capital employed (C). Thus, $E = ROC \times C$

Thus, we get: $EPV = \dfrac{E}{COC} = \dfrac{ROC}{COC} \ C$

Consider a business capitalized with $100m. If the COC is 10%, then the EPV >$100m, if and only if ROC > 10%.

## Growth

Now consider a special case of a firm growing indefinitely at a growth rate G. For simplicity, assume that revenues, capital employed and earnings are all growing at the same growth rate. That is, the ratios (net margin, ROC etc.) remain stable during this growth period.

Growth in earnings doesn’t come free. Part of the earnings (E) have to be reinvested (I) for growth, which results lowered cash flows (CF) that can be extracted from the business. $CF = E - I$

For earnings to grow by G*E, the amount of capital employed has to be increased by G*C, assuming ROC remains stable. This is the reinvestment required ( $I = G \times C$). Thus, $CF = E - I = ROC \times C - G \times C = (ROC - G) \ C$

The present value of a growing annuity is given by the Gordon Growth Formula: $PV = \dfrac{CF}{COC - G} = \dfrac{ROC - G}{COC - G} \ C$

Mathematically, if ROC > COC, growth enhances the value of the firm (PV > EPV).

## Value of Growth

One way to understand the value of growth is to consider the ratio of the growth scenario to the no-growth scenario. It is straightforward to show that: $M = \dfrac{PV}{EPV} = \dfrac{ROC - G}{COC - G} \ C \times \dfrac{COC}{ROC} \dfrac{1}{C}$

If we represent $g = G/COC$, and $r = ROC/COC$, then we have: $M = \dfrac{1 - (g/r)}{1 -g}$

If r > 1, this formula suggests reinvesting as much as possible (assuming incremental ROC = historic ROC) back into the business.

The figure below shows a contour plot of M(g, r). When r = 1 (ROC = COC), M = 1 regardless of growth. In this case, growth is neither good nor bad.

If r < 1, growth is actually bad. In this scenario, it might be best if the firm pays out all of its owner’s earnings. Divestment or -ve growth might also be shareholder friendly (see SPLS example). If r > 1, then growth is economically good.

### Example:

Consider a bank like Wells Fargo, with a historic ROC = 15%, a historic COC ~ 10%, and a growth rate of nearly equal to the COC (say G = 9% to keep the Gordon growth formula from blowing up).

Here g = 0.9, and r = 1.5.

The value of growth to WFC is  nearly a factor of 4.