Implied Growth: Solving for the Rate a Value Assumes

Implied growth is the terminal growth rate that makes a model hand back a value already in hand. Every other input is locked, that one is freed, and the equation is solved. On the invented case used here, Rs 2,128.14 crore needs 5.0000 per cent and Rs 2,240.00 crore needs 5.7976 per cent. Aswath Damodaran is named for reading a model backwards this way.

The whole technique rests on one lopsided fact about this model. Almost four fifths of the answer, 77.99 per cent of it, sits in the block that describes everything after the fifth forecast year, and the terminal growth rate is the only input that touches that block. A small movement in an input with that much leverage can absorb a large movement in the answer, so that input is the natural one to free. The other side of the same coin is the reinvestment coupling: growing at a given rate forever costs a fixed share of profit forever, so raising the growth rate raises the cost of that growth in the same movement, and the model does not permit one without the other.

What is the tool actually solving for?

Start with the shape of the thing. A discounted cash flow model for Sankalp Industrial Systems Limited, an invented manufacturer, is built in two blocks that are added together. The first block is five numbered forecast years, each with a free cash flow to the firmOnce a business has funded its own expansion, whatever it still holds belongs jointly to whoever lent to it and whoever holds its shares. attached, each discounted back at 12.00 per cent. The second block stands in for every year after the fifth, compressed into a single figure and then discounted back the same distance.

Run forwards, the model takes a terminal growth rate and produces an enterprise valueA figure for the operating business taken as a single unit, struck before any question of who takes which slice of it.. Run backwards, it takes a value and produces the terminal growth rate that would have produced it. Both directions are the same equation with a different unknown, and that is the entire idea. Nothing is added, nothing is estimated afresh, and no second model appears. All that has changed is which symbol is left empty.

Haggling for a second-hand tea stall on a busy corner is the same move. The owner wants Rs 4,00,000/- for it. Neither side can prove a price, so arguing about the price gets nowhere. Work out how many extra cups a day that price asks the buyer to believe in and the argument becomes an argument about cups. Both sides have actually counted cups. Implied growth is that move made on a model instead of a stall.

What does the equation say, term by term?

The whole model fits in one line, and no further term is ever added to it.

Because every input in that line except one is fixed by the record before the tool starts, the answer is a single number rather than a range. Free two symbols and the model has infinitely many solutions, because one can always be nudged to offset the other. Free one and it has exactly one, at least inside the range where the arithmetic behaves. Freeing exactly one symbol is not a nicety but the reason a reverse reading means anything at all.

The six steps below are the whole procedure, in order. Two later blocks point back at particular steps, so the steps are numbered.

One note on precision comes before anything is solved. The underlying record insists on it. Carried to the rupee, the five forecast years are worth Rs 4,68,41,43,564 and the whole model Rs 21,28,13,79,094. Every other figure here is rounded to the nearest lakh and written in crore, and on an answer that is 77.99 per cent closing block the last two digits of either rupee figure carry no information whatever. Two decimals of a crore is what is being rounded.

Step 4 is where most of the interest lies, and it is the step a simpler version of this model skips. Step 6 is mechanical: the calculator below walks it by bisectionHunting a number by halving the range it is known to sit inside, again and again, until the range is finer than the precision required for printing., halving a bracket until the value it produces and the value entered agree to more decimal places than anyone will ever print.

Why does faster growth cost cash before it pays any?

A stall serving two hundred cups a day cannot serve three hundred without a second urn, a second boy and a bigger gas cylinder. Growth is bought before it arrives. The same rule is wired into this model as arithmetic rather than as an opinion: if the business is to grow forever at a given rate, and each fresh rupee of capital earns 18.00 per cent forever, then it must put back that rate divided by 18.00 per cent of its profit forever. At 5.00 per cent that fraction is 5 over 18, being 27.78 per cent. At 6.00 per cent it is 6 over 18, being 33.33 per cent.

The fractions hide something, so follow the rupees instead. Take the rate up one point, to 6.00 per cent. Year 6 profit gains Rs 2.70 crore, reaching Rs 286.20 crore. The reinvestment the model must fund gains Rs 16.65 crore in that same movement, more than six times as much, so the cash left to value drops to Rs 190.80 crore.

The 5.00 and 6.00 per cent settings are not a special case. The reinvestment charge always climbs faster than the profit it is charged against, so the Year 6 cash flow on this model falls at every step of the growth rate from zero upwards, all the way to the ceiling. On this model, more terminal growth always means less terminal cash flow. The requirement that a closing block fund the expansion it claims belongs to Aswath Damodaran, and the arithmetic above is what carrying that requirement costs.

If the cash flow falls, why does the value go up?

Because two quantities shift together, and they shift against each other. The numerator of the terminal block shrinks by 6.81 per cent, as shown just above. The divisor shrinks as well, and by more. At 5.00 per cent growth the divisor is 12.00 less 5.00, being 7.00 percentage points, so each rupee of Year 6 cash flow turns into 14.2857 rupees of terminal value. At 6.00 per cent the divisor is 6.00 points, and each rupee turns into 16.6667. The capitalisation multiplier has risen 16.67 per cent.

The two movements multiplied together give the answer exactly: 0.9319 times 1.1667 is 1.0872, so the terminal value rises 8.72 per cent, from Rs 2,925.00 crore to Rs 3,180.00 crore. The value climbs not because the business throws off more cash but because the market is being asked to capitalise the same cash over a longer effective horizon. A reader who skips the coupling sees only the second movement and expects the value to jump by the full 16.67 per cent. The value jumps by roughly half that.

The coupling is also why the reinvestment-consistent version of the terminal block sits lower than the plain growing perpetuityAn income stream with no end date, growing at a steady rate while it runs. Valuing one collapses to a single division, and that is why it turns up wherever a forecast has to stop. most readers meet first. Growing the Year 5 cash flow of Rs 170.00 crore straight at 5.00 per cent and dividing by 7.00 points gives Rs 2,550.00 crore, and that route carries Year 5 reinvestment habits into perpetuity when the growth rate no longer needs them. Both constructions are standard. The reinvestment-consistent one is used throughout here, and the two are never mixed.

What shape does the answer trace?

Plot value against terminal growth across the whole reachable range and the curve does something a reader rarely predicts. Near the bottom it is almost a straight line and almost flat. Near the top it turns and goes very nearly vertical. The same one percentage point of terminal growth is worth Rs 144.69 crore at the bottom of the range and Rs 2,868.34 crore near the top. The second figure is 19.82 times the first, on the identical model with the identical five cash flows.

The reason is in the divisor and nowhere else. At 5.00 per cent the divisor is 7.00 points. At 10.00 per cent it is 2.00. At 11.00 per cent it is 1.00, and the same numerator has become a hundred times itself instead of fourteen times itself. Division by a number heading toward zero is the most violent arithmetic in ordinary finance, and it is sitting quietly in the middle of a formula everyone treats as routine.

The shape of the curve matters for reading a model as much as for building one. Two analysts can each be running a perfectly sound version of this equation, one anchored near 5.00 per cent and one anchored near 9.00 per cent, and the second will find every conversation about a tenth of a point far more heated than the first, and the difference lies nowhere in either analyst. The difference lies entirely in where each analyst is standing on the curve.

Read the last column downwards and the whole argument is there in one strip of figures. Nothing in the left column is unusual, nothing in the middle column is wrong, and the right column multiplies itself by twenty across nine rows.

What is ten basis points worth on this model?

A reader needs a unit before any of the percentages mean anything, and this is the unit. Near the base case, ten basis pointsOne per cent of one percentage point. Analysts count in them because a tenth of a point is awkward to say and easy to mishear. of terminal growth is worth about Rs 12.50 crore of value. Rs 12.50 crore is the honest rounded figure, and the two exact ones behind it are worth seeing. The ten points immediately above 5.00 per cent add Rs 12.69 crore, and the ten immediately below it add Rs 12.36 crore. The step is not even the same size on both sides of the base case. The curve's shape is showing up at the smallest scale anyone will ever look at it.

Carrying that Rs 12.50 crore up the ladder as though it were a constant understates every movement at the top of the range. Ten basis points above 7.00 per cent is worth Rs 24.20 crore. Ten basis points above 10.00 per cent is worth Rs 151.37 crore, or 11.93 times the figure at the base case. The unit belongs to the level at which it was measured, and to nowhere else on the curve.

What happens as the rate climbs toward the ceiling?

There is a hard limit on this arithmetic and it is not a modelling convention. What sits underneath the terminal value is the 12.00 per cent discount rate with the growth rate taken out of it, so the terminal growth rate has to stay below 12.00 per cent. Set the two equal and nothing is left underneath. Dividing by zero does not produce a very large number; it produces no number, and there is nothing for a tool to print.

As the growth rate approaches 12.00 per cent the value does not approach some enormous figure but no finite figure at all, and the honest behaviour is therefore refusal rather than a number with a warning attached to it. A tool that prints something vast and marks it approximate has quietly invented an answer where the arithmetic has none, and every downstream reader will treat the printed digits as real. The simulation below stays well inside the wall. Its widest value implies 8.3349 per cent and leaves 3.6651 points of divisor still standing.

The ladder, printed in full

The ladder below is static, and the control further down changes nothing in it. Every row was solved twice, once by halving a bracket and once by treating the reverse problem as the quadratic it actually is, and the two routes agree to ten decimal places.

Read the middle column downwards and notice how slowly it moves. The value in the first column climbs by Rs 836.00 crore from top to bottom, a rise of 44.00 per cent, and the rate it implies climbs by 5.3026 percentage points. The rate is the quiet column, and an eye trained on percentages underestimates it for exactly that reason. The last column, read downwards, shows the machinery that produces the effect: the divisor shrinks from 9.3876 points to 4.0850 while the numerator barely moves.

Where does a reader lose the plot?

The failure is almost never in the arithmetic. The failure is in the unit, and eighty basis points sounds like a rounding error.

How does anyone actually use a number like this?

An equity research associate updating a note has a model that says one thing and a screen that says another. Neither figure has a proof attached, so arguing about the difference in rupees goes nowhere. Running the value through this tool converts the argument into an argument about one assumption, and one assumption can at least be examined: what the long run would have to look like for the model and the observed figure to agree. The answer here is 5.7976 per cent instead of 5.00, and the associate writes exactly that and stops.

A credit reviewer sizing a facility uses it the other way round. Rather than asking what a value implies, the reviewer asks what a value would have to be for the assumption to look like the one already in the file, and reads the ladder upwards. A lending committee that has been carrying 5.00 per cent in its own long-run view can see, from the fourth row, what value that view supports on these cash flows, and can then discuss the gap in units it actually uses.

In both hands the tool converts an argument nobody can settle into an argument about a single assumption, and it does not settle that argument either. Someone haggling for a flat runs the identical move with none of the words for it: rather than saying the asking price is too high, they work out what the rent would have to be to justify it, and then everyone argues about rent, which both sides can observe. The technique is old and ordinary. The model adds precision about which assumption is carrying the weight.

What has to be held still for the answer to mean anything?

The rate this tool returns is a joint property of a model and a value. Change the model and the same value implies a different rate, without a rupee of the company's cash flow moving. The held inputs therefore belong on the screen beside the answer rather than in a note somewhere else, and a reverse reading quoted without them is close to meaningless.

Notice what is not on that list. There is no view about the company, no allowance for a control premium, no adjustment for a buyer with reasons of their own, and no judgement about the long run. The rate is arithmetic run backwards through a set of assumptions somebody else wrote down, and the tool has no opinion about any of them. Feed the same value into a model with a different weighted average cost of capitalOne annual rate standing in for what the whole of a company's funding costs it, weighted by how much of each kind of funding is in use., or mid-year discounting, or seven forecast years instead of five, and the implied rate comes out somewhere else entirely.

What does the tool refuse to do?

The tool refuses to say whether a rate it returns could be achieved. There is no setting in which it produces that, no threshold at which it starts warning, and no phrasing of the input that gets it there. The tool reports what a set of assumptions would have to contain in order to produce a value, and stops at the full stop.

The refusal is not modesty. The equation was handed a value, five cash flows, a rate and a return on new capital, and every one of those came from somewhere outside itself. A rate of 5.7976 per cent is what those inputs and that value jointly require. Whether the business behind them could sustain it forever is a question about industrial economics, competition and reinvestment opportunity, and no rearrangement of this formula contains a shred of evidence about any of them.

The second limit is quieter and matters as much. The calculator compares the chosen value with nothing. The calculator holds no reference figure, offers no verdict on the gap between two settings, and returns a rate for any value, including one nobody has ever quoted. A calculator that starts characterising its own output has stopped being a calculator, and this one prints rates and nothing else.

Sankalp Industrial Systems Limited is invented.
Educational material. Not advice on any investment, tax, budget or market position.