Scenario vs Sensitivity Analysis: Several Inputs or One

A sensitivity moves one assumption and holds the rest still. A scenario moves several together because the world links them. Moving Sankalp Industrial Systems Limited's cost of capital alone from 12.00 to 11.50 per cent gives Rs 2,302.69 crore. Moving four assumptions together gives Rs 2,626.89 crore. Adding the four separate effects lands at Rs 2,569.40 crore, and the Rs 57.49 crore missing is what counting inputs never reveals.

Everything below runs on one invented company, Sankalp Industrial Systems Limited, and departs from one starting point: a valuation of Rs 2,128.14 crore built on revenue rising Rs 120.00 crore a year, a 24.0 per cent margin on earnings before interest, tax, depreciation and amortisation (EBITDA)Earnings before interest, tax, depreciation and amortisation, taken as a share of the revenue that produced them., a 12.00 per cent cost of capital, a 5.00 per cent long-run growth rate and a 18.00 per cent return on new capital. The return on new capital never moves anywhere below. What the other four assumptions do stays visible without a fifth muddying it. Discounting is at each year end. Where those figures came from is settled elsewhere.

What is each of the two techniques actually for?

A home budget carries this same shape at a smaller size, so it is the place to begin. Suppose one salary of Rs 60,000 a month covers a household, and somebody wants to know how tight next year looks. One way to ask is: what if the rent goes up by Rs 3,000 and nothing else changes. The rent question is a clean one with an exact answer, and it can be asked four more times about school fees, the electricity bill, the commute and groceries. The result is five numbers, each of them true, each answering what that one line is worth.

The other way to ask is: what does next year look like if the salary earner changes jobs. A change of job is not one line moving. Those things are attached to each other through the job, so a new job usually moves the salary, the commute, the working hours and quite possibly the school the children go to, all at once. The first question holds everything still on purpose. Holding things still would describe a change that cannot actually happen that way, so the second question refuses to. Those are the two techniques, and they are confused constantly.

A sensitivity, defined on its own terms

A sensitivity moves exactly one assumption away from the starting point and holds every other assumption at its starting value. The result is exact. There is nothing approximate about it and nothing sloppy about it. A sensitivity answers a conditional question with complete precision: if this one input were different, and the model were otherwise untouched, the answer would be this.

The job a sensitivity does is attribution: working out how much of the answer is riding on each individual input. That job genuinely requires holding everything else still, because if two things move at once there is no longer any way to tell which of them moved the answer. Attribution is why a sensitivity is not a lazy or junior version of a scenario. A sensitivity answers a question a scenario structurally cannot answer, and a person who runs only scenarios never finds out which assumption their conclusion is actually resting on.

A scenario, defined on its own terms

A scenario moves several assumptions together, and the reason it moves them together is written down. Not a hunch, not a habit, not a rule that says three cases look thorough: a stated reason why these particular assumptions travel as a group. In the household example the reason is the job, and once the job is named the reason the salary and the commute and the school fees all moved is immediately visible.

The job a scenario does is to produce a number attached to a world somebody can describe out loud without contradicting themselves. That is a much stronger requirement than it sounds. The requirement rules out most combinations of assumptions, and the ones it rules out are exactly the ones that look most reasonable when they are arrived at one input at a time.

So is the difference simply how many inputs moved?

Here is the sentence that does the work, and it is not the one most people carry around. The difference between a sensitivity and a scenario is whether there is a stated reason for the assumptions to move together, and it is not the number of assumptions that moved.

The count is the test almost everybody reaches for, and the count gets it wrong in both directions. Two unrelated inputs moved together make a two-input sensitivity: a legitimate calculation with no story attached. Four inputs moved together with no reason connecting them make a four-input sensitivity. The four-input version is worse: the extra inputs make it look like a scenario without making it into one. Meanwhile a change of job in the household example might move only two things, the salary and the commute. The job is the reason, the reason is stateable, and the pair is a perfectly good scenario.

So the question to ask of any set of moved assumptions is not how many, but whether one sentence can be written explaining why these moved together, and whether that sentence survives being read out loud by someone who knows the business. If it does, it is a scenario. If it does not, several inputs have been moved at once and less has been learned than moving them one at a time would have taught.

The four sensitivities, computed

Take each of the four assumptions in turn, move it to its favourable setting, hold the other three at their starting values, and rebuild the valuation. Four calculations, four exact answers.

Every one of those four numbers is exact and every one answers a real question. Read as a group the four say something a scenario could never say. Half a percentage point off the rate is worth Rs 174.55 crore while a whole point on the margin is worth Rs 106.41 crore, so this valuation is riding harder on the cost of capital than on anything else, and by a wide margin. That is the attribution job, and it is done.

Why does the rate win so easily? The rate touches every rupee in the model, including the closing block that stands in for all the years after the fifth. The revenue increment and the margin touch only the cash flows, and the long-run growth rate touches only that closing block. A change worth fifty basis pointsOne hundredth of one percentage point, so fifty of them make half a point. on the rate is applied more widely than a change worth a full point anywhere else.

The combination that cannot happen

The third row of that table carries the lesson, so it repays a hard look. The cost of capital sensitivity says the valuation would be Rs 2,302.69 crore if the company were funded at 11.50 per cent instead of 12.00. Fine. A sensitivity holds everything still, and that is the point of it. So the other three assumptions in that row matter as much as the rate does.

The row holds revenue rising at Rs 120.00 crore a year. The margin stays at 24.0 per cent. The long-run growth rate stays at 5.00 per cent. So the sentence the arithmetic quietly commits to is this: a company whose funding became cheaper without its growth improving, without its margin improving, and without anyone expecting it to last any longer than before. Cheaper to fund, and better in no respect whatsoever.

Say that sentence out loud to somebody who lends money for a living and watch their face. Funding does not get cheaper for no reason. Funding gets cheaper because the business got safer, or grew, or paid down debt, or because the whole market repriced. Most of those things move at least one of the other three assumptions. The combination in that row is not impossible in the sense of breaking arithmetic. The combination is impossible in the sense that nobody can write two consistent sentences describing the company it refers to.

And that is fine, as long as it is clear which of the two things is in hand. As a calculation it is exact and useful, and it does the attribution job perfectly. As a description of a possible outcome it is nonsense, and the moment somebody puts it in a range and calls it the top end, it has been asked to do the second job while only being capable of the first.

If I run four sensitivities, have I built a scenario?

A one-at-a-time table invites exactly one thing from every reader who sees it: add the column up. Adding the column up is the most natural move in the world. Four effects, each carefully measured, so four effects together must be worth their sum.

Added, the four effects come to Rs 441.26 crore, and the valuation lands at Rs 2,569.40 crore. Run the four assumptions together in one model rather than four, the bull case, and the answer is Rs 2,626.89 crore. The gap of Rs 57.49 crore is not an error in either calculation, and it does not go away under a more careful check of the arithmetic. It is the interaction: the part of a combined effect that no one-at-a-time table can contain, because at no point while building that table are two of the changes present in the model at the same moment.

One rung where the printed cells disagree with the arithmetic

One detail in that addition deserves a stop. A careful reader will eventually flag it as an error, and it is not an error. The four printed effects added exactly as they appear in the table, Rs 93.20 plus Rs 106.41 plus Rs 174.55 plus Rs 67.11, come to Rs 441.27 crore. The same four effects computed without rounding any of them first, added, and rounded once at the end, come to Rs 441.26 crore. Three of the four cells happened to round upward, so the printed column overstates its own total by a paisa-scale amount that shows up at the second decimal.

The unrounded figure, Rs 441.26 crore, is the one carried throughout, and therefore an interaction of Rs 57.49 crore rather than Rs 57.48 crore. The rule behind that choice is worth more than the figure: round once, at the end, and never rebuild one number by subtracting or adding other numbers that have already been rounded for display. The unfavourable side of this same exercise is the opposite case, where the printed cells and the unrounded computation agree at Rs 545.60 crore, and it is precisely because the two cases cannot be told apart by looking that the rule has to be applied every time rather than checked once.

Where does the interaction actually come from?

One tempting explanation of the gap says the lower discount rate gets applied to cash flows that the higher revenue and the higher margin have already raised, so the effects multiply. The multiplying explanation is popular and, on this model, wrong in an instructive way.

Split the valuation into its two halves and measure the interaction in each. The five numbered years, the explicit forecast periodThe stretch of numbered years a model projects one at a time, before a single closing figure stands in for everything after., carry an interaction of minus Rs 2.37 crore. The closing block that stands in for everything after the fifth year carries plus Rs 59.87 crore. The whole of the interaction lives in the closing block, and the explicit years actually push a little the other way.

The reason the explicit years push the other way is a fact this case is built to expose: growing faster costs cash in the near term. A company adding Rs 150.00 crore of revenue a year has to fund more working capital and more plant than one adding Rs 120.00 crore, so its early free cash flow is lower, not higher. The explicit period is worth Rs 468.41 crore at the starting settings and only Rs 450.79 crore with all four favourable assumptions running. The cheap money is being applied to a smaller near-term stream, not a larger one.

So what is happening in the closing block? The closing block divides a cash flow by a gap. A stream with no end date is valued as a perpetuityA payment stream with no closing date, collapsed into one figure by dividing by a rate., so the two figures forming that gap are the funding cost on one side and, on the other, the terminal growth rateThe pace a model assumes a business keeps growing at once its numbered years run out, applied for good.. At the starting settings that gap is 12.00 less 5.00, or 7.00 points. Drop the rate to 11.50 and the gap becomes 6.50. Raise growth to 5.50 and the gap is also 6.50. Do both and the gap is 6.00.

Consider what that does to the multiplier the block divides by. One over 0.0700 is 14.2857. One over 0.0650 is 15.3846, so each change on its own is worth about 1.10 of multiplier. The two effects added predict 16.4835. But one over 0.0600 is 16.6667. A reciprocal bends, and the further the gap is compressed the more each additional point of compression is worth. So the two rate changes together are worth 0.1832 more of multiplier than the two changes measured separately. That is convexityThe property of a curve that bends, so equal steps along the input produce unequal steps in the output., and it is the single largest source of the interaction in this guide: of the Rs 57.49 crore, the cost of capital paired with the long-run growth rate accounts for Rs 18.12 crore on its own, more than any other pair.

Damodaran is the person to name for the rule that a perpetual growth rate has to be paid for out of the same block that claims it, and that rule is what plants the growth rate in two places at once here, above the line and below it. The doubling up is part of why this pair dominates. The full decomposition makes the mechanism checkable rather than assertable.

The interaction is positive here, and it is not positive everywhere

Both worked examples in this guide produce a positive interaction, so it is easy to walk away believing that running assumptions together always beats adding them. The belief is false, and it is worth showing why before it hardens.

The compressions and the expansions compound with each other, so the interaction is positive when the assumptions all push the same way. Move some favourably and others unfavourably at the same time and the sign flips. A compression of the closing gap from one assumption is partly undone by an expansion from another. Testing every corner of this model where at least one assumption moves against the others, ten of the fourteen mixed combinations produce a negative interaction, the largest of them at minus Rs 35.25 crore. The safe statement is that adding separate effects is wrong, not that it is wrong in a predictable direction, and any claim about the sign has to name the condition it holds under.

The same exercise on the unfavourable side

Run all four assumptions the other way and the arithmetic shows something the favourable side alone would have hidden.

The three anchors written out to the rupee

Everything above is printed in crore to two decimals, the right precision for reading. Underneath, the three cases each have an exact rupee value, and the reason for setting them out is that every derived figure in this guide was computed from these and not from the crore column.

The last digits carry no information whatever, and printing them is a statement about method rather than about precision. On a valuation where the block standing in for everything after year five does roughly three quarters of the work, a figure is not meaningful to the rupee and nobody should read it that way. The rupee column is for arithmetic: a crore figure rounded to two decimals cannot be multiplied back out to reach these, so any derived number built that way starts life a few rupees adrift and gets worse from there.

Adding the four unfavourable movements lands at Rs 1,582.54 crore. Running them together lands at Rs 1,654.94 crore. The additive version has taken Rs 72.40 crore too much off.

Here is the part worth being careful about. The direction of the miss is easy to state backwards. In raw value terms the additive answer sits below the run-together answer on both sides: Rs 57.49 crore below on the favourable side and Rs 72.40 crore below on the unfavourable side. The signed miss is in the same direction both times. The meaning of that miss flips once it is described as an effect. On the favourable side the additive version understates the rise; on the unfavourable side it overstates the fall; and because the miss is always downward in value terms, the whole additive range sits lower than the true one at both ends rather than being merely wider or narrower.

Checking one side therefore settles nothing about the other. The errors do not cancel and they are not symmetric. The two misses are Rs 57.49 crore and Rs 72.40 crore, one roughly a quarter larger than the other. An analyst who measures the favourable side, finds the miss modest, and assumes the unfavourable side behaves the same way has assumed something untrue about the size while being accidentally right about the sign.

What does a two-way grid establish, and what does it leave out?

The usual next move, once somebody has seen that one-at-a-time tables miss something, is to build a grid: the cost of capital along one edge, the long-run growth rate along the other, and a value in every cell. Six rate settings by five growth settings gives thirty cells, and thirty cells looks exhaustive.

The grid is a real improvement on a one-at-a-time table: within its two chosen assumptions it does capture the interaction. The cell where the rate is 11.50 per cent and growth is 5.50 per cent contains the Rs 18.12 crore of interaction between that pair, the largest pair on this model. A grid earns its place on that count alone.

A two-way grid varies two assumptions and pins everything else, however many cells it has. So every cell in that grid holds the revenue increment at Rs 120.00 crore a year and the margin at 24.0 per cent. So no cell in it is the bull case, which moves all four, and no cell is the bear case either. Its size is what makes this easy to forget: thirty cells feels like it has covered the ground, and the two assumptions it never touched are invisible precisely because they are the same in every cell.

The labelling rule, and it is three rules

Once the two can be told apart, naming them correctly stops being pedantry and starts protecting a reader who was not in the room.

The third row is the one that gets broken. Calling a one-at-a-time table a set of scenarios is the most common way to break it. Once the word scenario appears above those rows, a reader will treat each row as a world, because that is what the word means. Each row is not a world. Each row is a measurement of one input's leverage on the answer, and a measurement is not a forecast even when it is printed in rupees.

How to tell, from the outside, which technique produced a range

Other people's ranges turn up far more often than ranges built in-house, and the document usually does not say which technique made it. Four checks settle it quickly, and none of them needs the model.

First, whether the two ends can be described. If somebody can say in two sentences what kind of company sits at the top end and what kind sits at the bottom, and the two descriptions differ in more than one respect, it came from scenarios. If the only available description of the top end is the name of one input, it came from a sensitivity.

Second, count what moved at each end, then check whether the same things moved. Sensitivity ends move different inputs from each other, and that is the tell. Both ends of a scenario move the same inputs, in opposite directions.

Third, look at how the range sits around the starting point. A range built from the extremes of one-input runs is bounded by whichever single input is strongest, so it hugs the starting point unevenly: here Rs 174.55 crore above and Rs 159.61 crore below. Four assumptions push at once in a properly run pair of cases, and their interaction is included, so the pair opens much wider: here Rs 498.75 crore above and Rs 473.20 crore below.

Fourth, ask whether the ends were added or run. If the top end equals the starting point plus a sum of separately measured effects, it was added, and it is understating by exactly the interaction it could not see. On this company that test has a signature checkable by eye: an added favourable end lands on Rs 2,569.40 crore while a run one lands on Rs 2,626.89 crore.

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