Sensitivity and scenario analysis: what a single valuation hides
A valuation gives one number, and every input behind it is a guess. Sensitivity analysis moves one input at a time to see which guesses matter. Scenario analysis moves several together into a worst, a base and a best case. This lesson does both on the Apple valuation from the previous module and ends with a result that is easy to miss: even the best case can sit below the market price.
- Level
- Beginner
- Reading time
- About 20 minutes
- Know this first
- What a share price implies
By the end you will be able to:
- move one input at a time and rank the inputs by how much they change the answer
- build a worst, base and best case and weight them by probability
- say why the probability-weighted value is not the same as the base case
- read a range of values against a market price without treating either end as a forecast
One number, many guesses
The previous module ended with a valuation of Apple: a stream of after-tax operating profit, a share of it reinvested, a discount rate, and a value for the years beyond the forecast. The answer came out as a single figure, and a single figure looks more certain than it is. Every input behind it is a judgement about the future. Change a few of them and the figure changes, sometimes a great deal.
Two simple habits deal with this. Sensitivity analysis moves one input at a time while the others stay put, to find out which guesses matter and which hardly do. Scenario analysis moves several inputs together into a small number of stories, usually a worst case, a base case and a best case, and often weights them by how likely each seems. Neither produces a better forecast. Both show how wide the range of reasonable answers is, and that is what a reader needs to know before trusting any one of them.
This lesson uses the Apple valuation from the discounted cash flow lesson, so the starting point is familiar: after-tax operating profit of $112.3 billion in fiscal 2025, net cash of $33.76 billion and 14.776 billion shares. The base case below assumes profit grows 5% a year for five years, a discount rate of 8.9%, growth of 3% for ever after that, 10% of profit reinvested in the first five years and a 20% return on new investment afterwards. The 5% is a choice made for this lesson, not a forecast. It is below the 8.0% growth in Apple's operating income in the latest year, so it is neither generous nor harsh, and that is all that is claimed for it. On those inputs one Apple share is worth $127.
Moving one input at a time
The method is plain. Keep the base case, change one input to a lower and a higher value, and record the value per share each time. The gap between the two results, the swing, measures how much that one input matters over the range tried. The table does this for five inputs.
| Input changed | Range tried | Value per share at the first end | Value per share at the second end | Swing |
|---|---|---|---|---|
| Growth in profit, years 1 to 5 | 0% to 10% | $103 | $156 | $54 |
| Discount rate | 9.9% to 7.9% | $109 | $152 | $43 |
| Growth after year 5 | 2% to 4% | $117 | $140 | $23 |
| Return on new investment after year 5 | 10% to 30% | $110 | $132 | $22 |
| Profit reinvested, years 1 to 5 | 20% to 5% | $124 | $129 | $5 |
Over these ranges, growth in the first five years moves the answer most, by $54 a share, followed by the discount rate at $43, growth after year 5 at $23 and the return on new investment at $22. The share of profit reinvested in the first five years moves it least, by $5, because it only changes how much cash is taken out of the business in those years and leaves the later years alone.
Two cautions go with any such ranking. First, it depends entirely on the ranges chosen. A discount rate tried between 7.9% and 9.9% is a range of two points; growth tried between 0% and 10% is a range of ten points. Had the rate been tried over a wider range, it would have ranked higher. The ranking says which inputs matter for the ranges a reader finds plausible, so the ranges themselves deserve as much thought as the answer. Second, the inputs are not independent in practice. A company that grows faster usually needs to reinvest more, and a change in interest rates moves the discount rate and the long-run growth rate together. Moving one input alone is a way to learn how the model behaves, not a description of what the world might do.
Two inputs together
When two inputs matter most, a grid shows both at once. The table below crosses the discount rate with five-year growth, holding everything else at the base case.
| Discount rate | Growth 0% | Growth 5% | Growth 10% |
|---|---|---|---|
| 7.9% | $122 | $152 | $188 |
| 8.9% | $103 | $127 | $156 |
| 9.9% | $88 | $109 | $133 |
The grid shows how the two trade against each other. With no growth at all, a discount rate one point lower gives $122 a share, close to the base-case $127: a point off the rate buys about as much as five points of yearly growth. The corners are far apart, from $88 with a 9.9% rate and no growth to $188 with a 7.9% rate and 10% growth. Both corners use inputs that someone could defend, and that is the main thing a sensitivity grid teaches.
Three stories instead of one
A grid cannot say which combination to believe. A scenario can. Each scenario is a short story told through the inputs: in the worst case growth is weak and investors demand a high return, in the best case growth is strong and the required return is lower, and the base case sits between. Each gets a weight, the probability the analyst gives it, and the probability-weighted value is the sum of each value multiplied by its weight.
The weights are guesses too, and they should be written down so that a reader can change them. The ledger below uses a worst case of $80 a share, a base case of $127 and a best case of $184 at the starting settings. The calculator lets the reader change the growth and the discount rate in the worst and the best cases, and the weights, to see how the result responds.
| Case | Growth, years 1 to 5 | Discount rate | Weight | Value per share | Against the market price |
|---|---|---|---|---|---|
| Worst | -2% | 10% | 25% | $80 | −64% |
| Base | 5% | 8.9% | 50% | $127 | −42% |
| Best | 10% | 8% | 25% | $184 | −16% |
Build your own three cases
The left page of the ledger shows the value per share in each case. The right page weighs them and sets the result against Apple's market value per share of $220.2, which is the $3,253.4 billion market value of shares held by non-affiliates at 28 March 2025 divided by the 14.776 billion shares outstanding in October 2025. Those are two different dates, so the per-share figure is a rough marker for comparison, not a quotation. The weight on the base case is whatever is left after the other two, so the three always add to 100%.
Worst case $80, base case $127, best case $184 a share. Weighting them 25% / 50% / 25% gives $130, which is 41.2% below the market value of $220 a share. The gap between the best and worst case is $104.
Four experiments. First, at the starting settings the weighted value is $130, which is 41.2% below the market value per share. Second, move the weights to 10% worst, 10% best and 80% base: the weighted value falls to $128, close to the base case, because most of the weight sits there. Third, give the best case 20% growth at an 8% rate: that single case is worth $275, above the market value per share, which shows how much has to be assumed to get there. Fourth, give the worst case a decline of 10% a year at a 12% rate and it falls to $46. The scale of the swings is the point: the range between best and worst is $104 a share at the starting settings, about 82% of the base value itself.
Why the weighted value is not the base case
At the starting settings the weighted value of $130 is $2.6 above the base case of $127, though the weights on the worst and best cases are equal. The reason is that the cases are not equally far from the base: the best case is $57.3 above it and the worst is only $46.8 below. Higher growth and a lower discount rate compound together, so good outcomes add more value than bad ones take away. When a range is lopsided like this, a base case that sits in the middle of the assumptions is not in the middle of the values, and the weighted value is the better single summary.
The result also works in the other direction. If the worst case were harsher than the best case is generous, the weighted value would fall below the base case. The only way to know is to compute it, which is why the calculator weights the three cases rather than leaving the sum to the reader.
What the range says about the price
Even the best case of $184 a share sits below the market value per share of $220. Reading that correctly takes some care. It does not show that the market is wrong, because the cases are the author's: growth of 10% a year for five years and an 8% discount rate were picked for this lesson. What it shows is how much the market price asks for. With the base-case discount rate of 8.9%, five-year growth of 18.7% a year is needed to reach the market value per share, against 8.0% for Apple's latest operating income growth. At the 8.0% rate the value is $144 a share. The reverse valuation in the previous lesson reached the same conclusion from the other direction.
A reader who thinks the price is fair has to believe in a case better than the best one here, or a lower discount rate, or growth that lasts longer than five years. A reader who thinks it is too high has to believe the opposite. Scenario analysis does not settle that argument. It makes both sides say what they are assuming.
What these methods cannot do
- They do not make the inputs better. A careful range around a poor base case is still a poor analysis.
- The cases are chosen by the analyst. A worst case that is not very bad, or a best case that is not very good, makes the range look narrower than it should be.
- The weights are guesses. A probability of 25% sounds precise and is not.
- Real outcomes can fall outside every case. Three cases are a way of thinking about a range, not a list of what can happen.
- A single swing hides interactions. The one-at-a-time table cannot show that two inputs moving together matter more, or less, than the sum of the parts. The grid shows it for two inputs and no further.
Common mistakes
Treating the base case as the forecast
The base case is the middle of the assumptions chosen, not a prediction. Present it with the range around it.
Ranking inputs without looking at the ranges
An input looks unimportant if it was tried over a narrow range and important if it was tried over a wide one. Say what ranges were used.
Making the worst case mild
A worst case that is only a little worse than the base gives a false sense of safety. Ask what would have to go wrong for the business to be in real trouble, and use that.
Reading probabilities as facts
Weights of 25%, 50% and 25% are a way to combine three views. They are not measured frequencies.
Averaging the inputs instead of the values
The weighted value is the sum of the values times their weights. Plugging the weighted average of the inputs into the model gives a different answer, because the value does not move in a straight line with the inputs.
Try it yourself
1. A share is worth $80 in the worst case, $120 in the base case and $200 in the best case. The weights are 25%, 50% and 25%. What is the weighted value?
Show the working
0.25 × 80 + 0.50 × 120 + 0.25 × 200 = 20 + 60 + 50 = $130.
2. Why is the answer to the first question above the base case of $120, even though the worst and best cases are weighted equally?
Show the working
The best case is $80 above the base and the worst is only $40 below it. Equal weights on an uneven spread pull the average up.
3. A table shows that a one-point change in the discount rate moves the value by $18 and a one-point change in long-run growth moves it by $12. Does that mean the discount rate matters more?
Show the working
Only for those ranges. If the discount rate is known within a quarter of a point and long-run growth could be anywhere across four points, the growth rate may cause more of the uncertainty in the answer. The question to ask is how far each input could plausibly be wrong.
4. The best case comes to $184 and the market value per share is $220. List two things a buyer at $220 might be assuming.
Show the working
A lower discount rate than the cases use, or growth that is stronger or lasts longer than the best case allows. Each is a claim that can be examined, and none of them shows the buyer is wrong.
Where this is used
- An analyst puts a range around a target price instead of a single figure.
- A company tests a large investment against a downside case before committing to it.
- A lender asks how much sales or margin can fall before a borrower cannot meet its payments.
- A board is shown a plan with a base, a downside and an upside, and asked which one the budget rests on.
Key terms
- Sensitivity analysis
- Changing one input at a time, with the rest held fixed, to see how much the answer moves.
- Swing
- The difference between the results at the two ends of the range tried for one input.
- Scenario analysis
- Changing several inputs together into a small number of coherent cases.
- Base case
- The set of assumptions the analyst treats as the central view.
- Worst and best case
- Plausible cases at the unfavourable and favourable ends of the range, not the absolute limits.
- Probability-weighted value
- The sum of each case's value times the weight given to that case.
Questions people ask
How many scenarios should there be?
Three is the usual number because it is easy to explain. More cases add detail but also more guesses, and a reader cannot easily keep track of them.
Should the base case get the most weight?
Usually, but nothing requires it. The weights say how much each story is believed, and an analyst who thinks outcomes are likely to be unusual can say so.
Is this the same as a Monte Carlo simulation?
No. A simulation draws thousands of random combinations of inputs from stated ranges. Scenario analysis picks a few by hand. The simulation covers more ground, and it still depends on the ranges and the relationships chosen at the start.
Does a wide range mean the business is risky?
It means the value is sensitive to the inputs, which can be because of the business or because of how little is known about it. The lesson on the cost of capital covers risk in the discount rate itself.
Sources and review
- Apple Inc., Form 8-K, Exhibit 99.1: fourth quarter and full-year fiscal 2025 results, filed with the US Securities and Exchange Commission. Source of the income statement, balance sheet and cash flow statement figures.
- Apple Inc., Form 10-K for the fiscal year ended 27 September 2025. Source of the market value of shares held by non-affiliates at 28 March 2025.
How the numbers were checked: the starting profit, net cash, share count and market value are the Apple figures used in the valuation module. Each value per share in the lesson, every cell of the tables and the weighted values were recomputed in plain code, and the calculator, which runs separately, was compared with that code across a grid of inputs.