Two-Stage vs Three-Stage DCF: Where the Growth Step Goes

A two-stage discounted cash flow (DCF) model writes out a handful of years and then hands everything after them to one perpetuity, so growth and reinvestment both jump at the seam. On Sankalp Industrial Systems Limited, an invented manufacturer, growth drops 2.14 points and reinvestment 9.26 points in a single instant. A three-stage model inserts a fading period so that both quantities travel instead of jumping.

Underneath the argument sits one question about businesses rather than about spreadsheets: how does a company get from doing unusually well to being ordinary? Something has to carry it there. A two-stage model has nowhere to put that journey except a seam, and a seam has no duration. The alternative says the journey is itself a period of time, with a length and a shape that can be argued about. Every other difference between the two shapes falls out of that one disagreement about whether becoming ordinary takes any time at all.

Think about a roadside snack cart that has just found a busy corner. For two or three years it takes more money every month, and the owner keeps buying another burner, another counter, another cart. Then the corner fills up. There is a version of that story where the good years simply stop on a particular Tuesday, and a version where they thin out slowly over four monsoons. Nobody arguing about a valuation model is arguing about anything else.

What are the two stages, and what does each one hold?

Stage one is the explicit forecast periodYears set out one at a time on a sheet. One formula handles everything that follows them.. On Sankalp Industrial Systems Limited it runs five years and every line in it is written out by hand. Revenue moves from Rs 12,00,00,00,000 in the last completed year to Rs 18,00,00,00,000 in Year 5, adding exactly Rs 1,20,00,00,000 each year. Because the rupee increment never changes while the base it lands on keeps rising, the growth rate falls of its own accord: 10.00, then 9.09, then 8.33, then 7.69, then 7.14 per cent.

The same five years also carry a reinvestment story. The company puts back Rs 1,00,00,00,000 of net new capital every year, and its net operating profit after tax (NOPAT)Profit from operations, less the tax charged on it, with nothing at all removed for interest. Lenders and shareholders share the figure between them. starts the forecast at Rs 1,98,00,00,000 and ends it at Rs 2,70,00,00,000. A numerator that never moves, over a denominator that keeps rising, falls, so the reinvestment rate walks down beside the growth rate: 50.51, 46.30, 42.74, 39.68 and 37.04 per cent. Both of the quantities that will jump at the seam are, inside stage one, already declining smoothly and by themselves.

Stage two is one line of arithmetic covering the rest of time. Everything from Year 6 is assumed to expand at 5.00 per cent a year in nominalCounted in the rupees of the day, with nothing taken out for what prices have done in the meantime. rupees, forever, on a reinvestment rate of 5 over 18, or 27.78 per cent. Year 6 operating profit after tax is Rs 2,83,50,00,000, the capital put back is Rs 78,75,00,000, and the cash left is Rs 2,04,75,00,000. Capitalised at the 12.00 per cent cost of capital this worked example assumes, less the 5.00 per cent growth, that is a terminal value of Rs 29,25,00,00,000.

Between the two stages there is nothing. Not a short period, not a transitional year: an instant. Year 5 ends and Year 6 begins, and the business is assumed to have arrived at a steady stateThe condition a business is taken to have reached once nothing about its growth or its spending is expected to change again. on the way across.

What does a three-stage model put where that instant was?

A three-stage model keeps stage one exactly as it is and keeps the perpetuity exactly as it is. Between them it inserts a fading period, and everything the extra stage does happens inside that inserted block. Growth walks down from 7.14 per cent towards 5.00 per cent across some number of years instead of arriving there in one instant. Reinvestment walks down from 37.04 per cent towards 27.78 per cent beside it, staying tied to growth in every intermediate year rather than only at the two ends. The perpetuity then starts at the far edge of the fade instead of at the close of Year 5.

Nothing else about the model changes. The two shapes are the same forecast, the same rate and the same perpetuity, and one inserted period is the whole of the difference. Most descriptions of the three-stage model make it sound like a different technique, and it is not. A three-stage model is the identical technique with a block of years wedged into the middle, and the argument is entirely about whether that block belongs there.

How far does growth actually fall at the seam?

Now the two shapes can be held against each other, and the honest measure is the discontinuity the third stage exists to remove. Revenue in the final forecast year of this model expands 7.14 per cent. Everything after it expands 5.00 per cent. The drop of 2.14 percentage points between those two readings happens at a single instant.

Draw the whole growth path on one axis and the shape of the objection becomes obvious. Through the forecast the line descends in easy steps of roughly seven tenths of a point at a time. Then it drops more than three of those steps at once, with no year in between to hold the movement. The criticism of a two-stage model is not that 5.00 per cent is the wrong terminal rate; it is that the path to 5.00 per cent has no width.

There is a second step at the same instant. How big is that one?

Reinvestment jumps too, and almost nobody says so. In Year 5, out of every rupee of operating profit after tax, 37.04 paise never leaves the business at all. From Year 6 the figure is 27.78 paise. The reinvestment step is 9.26 percentage points, gone at the same instant the growth rate loses 2.14. The step nobody mentions outweighs the step everybody argues about by better than four to one.

Stated plainly: one decision at the seam, taken about growth, produces two discontinuities, and the larger of the two is in a row most readers never look at.

Why does one decision produce two steps rather than one?

Because the two are chained to each other. Capital put back into a business is what makes the business larger, and how much larger depends on what that capital earns once it is installed. Koller, Goedhart and Wessels write value, expansion and the return earned on capital as a single expression, and the value driver formulaOne expression holding value, the rate of expansion and the return capital earns. Settle any two of the three and the last is settled. is that expression doing its work at the seam.

On this forecast the return on newly invested capital is 18.00 per cent, and that is an assumption rather than a fact about the company. Hold it fixed and the arithmetic closes with no room left. Year 5 holds back 37.04 per cent, and 37.04 scaled by the 18.00 pays for 6.67 per cent. The terminal period books 5.00 per cent, so 5.00 over the 18.00 fixes its reinvestment at 27.78 per cent. Once the modeller decides that growth falls to 5.00 per cent, the reinvestment rate is no longer available to choose. One decision therefore lands as two steps.

Run the check on the forecast years too. They were built so it would close. Four rows, four exact agreements, and the same chain that holds in each of them is what forces the second step at the seam.

What must a fading period hold true in every one of its years?

The same tie, in every single row. The tie is the real discipline the third stage imposes, and the tie is what most three-stage models get wrong. Inside a fade there is no such thing as a growth assumption on its own. What fresh capital earns converts either quantity into the other, and it sits there whether the sheet references it or not. So picking the growth for the fourth year of the fade picks the reinvestment for that same year.

The figures already established make this concrete. On the way down from 37.04 per cent to 27.78 per cent, a fade year booking 6.00 per cent must hand back 6.00 divided by 18.00 of its profit, or 33.33 per cent. A fade year booking 5.50 per cent must be handing back 30.56 per cent. Neither figure is a matter of judgement once the growth for that year is chosen and the 18.00 per cent is held. A fading period is not one new row on the sheet but two, and the second one is derived from the first rather than typed.

Notice the consequence for the model as a whole. A two-stage model resolves the tie once, at the seam, in full view. A three-stage model has to resolve it in every intermediate year, quietly, in rows nobody reads closely. The extra stage does not remove the discipline. Instead it multiplies the number of places the discipline has to be applied, and a fade is therefore either wired together with a formula or not worth having.

Does the cash flow step down at the seam as well?

Cash flow rises, and the rise is the part of the argument most worth sitting with. Year 5 free cash flow is Rs 1,70,00,00,000. Year 6 free cash flow is Rs 2,04,75,00,000. The difference is Rs 34,75,00,000, a rise of 20.44 per cent, at the very instant two assumptions jumped.

The rise decomposes cleanly into the two steps pulling opposite ways. Growing Year 5 profit at 5.00 per cent adds Rs 13,50,00,000 of operating profit after tax. The reinvestment bill was Rs 1,00,00,00,000 a year and becomes Rs 78,75,00,000, so cutting it hands back another Rs 21,25,00,000. The two additions come to the Rs 34,75,00,000 exactly. The growth step reduced how fast profit climbs; the reinvestment step reduced how much of that profit is spent, and on this model the second effect is comfortably the bigger of the two in the first terminal year.

A reader watching only the cash flow row would see a line rising smoothly through the seam and would conclude, wrongly, that there is no discontinuity there at all. The two rates therefore have to be inspected rather than the cash. Cash is the output of the model, and outputs are where discontinuities go to hide; the assumptions are where they live.

Does a third stage change the answer, or only the shape?

A fading period does one thing to the value: it defers the perpetuity by the length of the fade, and it changes what enters the perpetuity when the model finally gets there. A fading period never removes the perpetuity. A fair number of modellers reach for a third stage hoping the terminal block will stop dominating, and the arithmetic gives them very little.

One comparison sizes deferral without inventing a fade at all. Take the terminal cash flow of Rs 2,04,75,00,000 and treat it as a five year annuityA run of equal payments that stops on a stated date. Only the stopping separates it from a stream with no end. at the 12.00 per cent rate this example assumes: it is worth Rs 7,38,07,79,274. Treat the same cash as a growing perpetuity at 5.00 per cent and it is worth Rs 29,25,00,00,000. The perpetuity is 3.96 times the annuity, and the difference between the two treatments is Rs 21,86,92,20,726.

Five years of a cash stream are worth about a quarter of the same stream running for ever, so pushing the perpetuity back by a few years moves far less value than the change to the picture suggests. Read the whole two-stage build the same way and the point sharpens further. Of the answer it produces, 77.99 per cent sits in the terminal block. Inserting years in front of that block changes which arithmetic produces the value; it does not change where the value is.

There is a second effect, and it runs the other way. The net movement from a fade is genuinely hard to sign in advance. A fade does not only delay the perpetuity: it also changes the cash that walks into it. Years spent growing above 5.00 per cent leave the business bigger when the perpetuity finally starts, so the profit being capitalised at the far end is larger than the Rs 2,83,50,00,000 a two-stage build capitalises. Deferral pulls the answer down and a bigger base pushes it up, and which one wins depends entirely on the fade length and the fade path nobody has recorded. The direction of the change is not settled, let alone its size.

What does the extra stage cost?

Two assumptions, and both of them are invisible in the output. The first is a fade length. Five years, ten, fifteen: nothing in a set of accounts, a filing or a management commentary tells anybody which. The second is a fade path. Straight line, front loaded, back loaded, or some curve fitted to taste. Nothing tells anybody that either.

A two-stage join carries one unobservable assumption and a three-stage join carries three, and the extra two are harder to interrogate than the one they were brought in to soften. The trade is not obviously a good one. A two-stage model puts its single unchecked assumption in the open, in a spot everybody knows to attack. A three-stage model spreads three of them across a block of rows that looks exactly like a forecast, and rows that look like forecasts get read as though somebody measured them.

Think of a household deciding how long a salary increase will keep coming. Saying it stops next April is crude, and everybody can see the crudeness and argue about the date. Saying it tapers gently over seven years and drawing a curve for the taper feels more careful, and it is more careful only if somebody can say why seven and why that curve. Otherwise it is the same guess wearing a nicer suit.

When does the extra stage earn its place?

One question decides it: does this business have a transition that can actually be described? A patent running to a date on the register. A contract book unwinding over a term somebody has written down. A capacity build with a commissioning schedule attached to it. In each of those the modeller can say something specific about how long the change takes and what shape it has, and a fade is then an expression of knowledge rather than a decoration on top of ignorance.

A company that simply grows faster than average, for no reason anybody has named, has no describable transition, and a fade fitted to it is a line drawn between two guesses. Smoothness for its own sake is not evidence of care. Smoothness is the appearance of care, and the appearance is harder to challenge than plain crudeness.

Test it against this example honestly. This forecast names no patent, no unwinding contract book and no commissioning ramp for Sankalp Industrial Systems Limited. The forecast does name a rate of return on the capital in the business: 15.00 per cent in the last completed year, and 15.88 per cent by Year 5. The return climbs because fresh capital is assumed to do better than what is already installed, 18.00 against 15.00 per cent. Invested capital ends the five years at Rs 17,00,00,00,000 having started at Rs 12,00,00,00,000, and profit outruns that. The company is not converging on ordinariness inside the forecast; it is drifting the other way.

None of that argues for either shape; it is a property of this particular example. A rising return is tempting to read as proof that a fade is needed, but a fade needs a transition somebody can describe, not merely a number moving in an awkward direction. The rise establishes only that the business has not become ordinary by Year 5, and the length of the written out forecast is a separate subject.

One more thing sits under the 18.00 per cent, and naming it makes the assumption much easier to attack. Margins in this forecast never move. So the 18 against 15 has nothing whatever to do with margins. The 18 against 15 is really about capital turnoverSales measured against the capital standing behind them, counted as rupees earned for each rupee tied up., and the two rows below say the whole of it. Fresh capacity is assumed to spin faster than installed capacity, and nothing in the record supports that.

What is fixed in this example, and what cannot be computed?

The two-stage figures above all come from one computed model: the five forecast years, the terminal build, the Rs 29,25,00,00,000 terminal value and the Rs 21,28,13,79,094 enterprise value. Each of those is fixed by the assumptions stated beside it.

Neither a fade length nor a fade path can be read off the accounts of Sankalp Industrial Systems Limited, so no three-stage valuation of it can be computed. The effect of a third stage and the cost of a third stage are both exact. A three-stage figure for this company is not: producing one would mean inventing two assumptions in order to manufacture a third.

An illustrative three-stage figure would be quoted somewhere else next month as though somebody had computed it, stripped of the caveat that invented it to show a shape. Numbers travel much better than the caveats attached to them. The comparison stands without one. The deferral arithmetic already sizes what a fade moves, and the assumption count already sizes what it costs.

A fade drawn from 7.14 per cent down to 5.00 per cent would need a length and a path before it could be drawn at all, and it would then hand over a three-stage valuation of this company that nobody had computed. Wrapping that invention in a moving control would not make it any less of an invention. The terminal growth rate and the number of years written out are the two relationships that can honestly be moved, and both are settled under separate subjects.