What the price already assumes

A forward DCF invites an argument about your growth guess, and you will usually win it with yourself. A reverse DCF changes the question. Given a price, the model hands back the growth the price needs, and now the burden sits on that number: does this business look like one that compounds owner earnings at that rate for ten years? You can disagree with a price without pretending to know the future. You only have to find its implied growth implausible.

Two cautions keep the tool honest. First, the implied growth is only as meaningful as the three inputs you held fixed; change the discount rate and the answer moves, which the self-study card in Part 3 works through. Second, the result inherits every label on the base: here, owner earnings by the all-capex approximation, averaged over ten years, from 10-K facts (Analyst sessions 1 and 2).

There is no formula that inverts a two-stage DCF for growth, so we search. The search works because the value only ever rises as stage-one growth rises: every year's cash is bigger, and so is the terminal cash. A function that only rises crosses any price at most once. Pick a low growth whose value sits below the price and a high growth whose value sits above it, try the middle, and keep whichever half still has the price between its ends.

Each try halves the bracket. Starting from -50 to 100 percent, a width of 1.5, ten halvings leave a bracket about 0.0015 wide, a fifth of a percentage point; around sixty leave it narrower than the gap between neighbouring double-precision numbers near 14 percent. Two hundred is far more than needed, and it is cheap: two hundred calls of a ten-year loop. The guard matters more than the count. If the typed price sits below the value at -50 percent, or above the value at 100 percent, there is no answer inside the bracket, and the function must say so instead of returning an edge.

14.17 percent a year for ten years is what a test price of twice the value needs, with everything after year 10 still growing at 2.5 percent and still discounted at 9 percent. It means owner earnings per share roughly three and three-quarter times larger by year 10. That is the claim; whether it is plausible is Part 3.

-4.31 percent a year is what half the value needs: ten straight years of shrinking owner earnings before settling into 2.5 percent growth. A price that low is not saying "no growth". It is saying "decline, for a decade".

Notice the asymmetry in the curve. Going from 1x to 2x adds 9.17 points of growth; going from 1x to 0.5x takes away 9.31. Because so much of the value sits beyond year 10 (57.0 percent at these defaults, Analyst session 4), stage-one growth has to move a long way to move the total.

Every term in the DCF is the base multiplied by something that depends only on r, n, g and gt. Double the base and the value doubles; so does the test price at "twice the value". The ratio between price and value is all the reverse function ever sees. That is why Caterpillar's rung-3 value of 250.78, Exxon Mobil's 117.34, J&J's 150.81 and P&G's 95.16 all give 14.17 percent at twice their value and -4.31 percent at half, at the same defaults.

The practical point: implied growth depends on the price-to-value ratio and on the three fixed inputs, not on which company or which base. Change the base and you change the value, so a given price implies a different growth. Change the discount rate and you change the mapping itself.

The base rates. Across the top 1,000 global companies by market value since 1950, the Base Rate Book (read at source, hosted copy) finds three-year sales growth with a mean of 8.1 percent and a median of 5.4 percent a year, inflation-adjusted, with a standard deviation of 18.7 percent. 23 percent of companies had negative real sales growth over three years. Growth rates, and their spread, decline as companies get bigger: the four filers here are all large.

The filers' own history. Mean owner earnings per share over each run's first five years against its last five: Caterpillar 9.2 percent a year, Exxon Mobil 8.1, J&J 7.3, P&G 3.1. These are nominal, so they include inflation, while the base rates are real; and they are owner earnings per share, not sales. The two must not be compared digit for digit. What survives the mismatch is direction: 14.17 percent for ten years sits above every filer's own nominal record and well above the real median for companies of any size.

None of that says 14.17 percent is impossible. It says a test price of twice the value is a bet on an outcome these businesses have not produced, and the person making it should be able to name why the next decade differs from the last.

Hold the test price at 190.33 and raise the discount rate from 9 to 10 percent. Every year's cash is now worth less today, so the same price needs more growth to justify it: implied growth rises. Lower the rate to 8 percent and implied growth falls. The reverse DCF did not remove the discount rate judgment from Analyst session 4; it kept it, quietly, inside the answer.

So an implied growth figure travels with its rate, just as a value does: "190.33 implies 14.17 percent a year for ten years at 9 percent, 2.5 percent terminal". Run reverse() at 8 and 10 percent yourself in homework 2 and print the two answers; the size of the swing is the honest width of the claim. Analyst session 6 builds the full grid of which input moves the answer most.