Lesson 4 of 6
The sensitivity is the answer
Nine cells from the same nine filings, a range of 96.56 to 136.60 dollars a share, and why the grid rather than the point is what the model produced.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
Who writes thisEditorial processLinkedIn (opens in a new tab)X (opens in a new tab)
In one minute
- Nine cells built from the same nine filings span 96.56 to 136.60 dollars a share, a range of 41.5 percent of the lower figure.
- The grid is the output of the model. The single cell in the middle is a convenience, not a finding.
- Choosing the cell you prefer after seeing the grid is how a model becomes an argument for something already decided.
The previous lesson produced one number. This one produces nine, from exactly the same filings, and the nine are the honest output.
Value per share across the two choices that move it most
- 11% return, 1.5% growth96.56$
- 11% return, 2.5% growth104.54$
- 10% return, 2.0% growth112.87$
- 9% return, 1.5% growth122.40$
- 9% return, 2.5% growth136.60$
The middle cell, $112.87: 112.87$
The grid
Rows are the required return. Columns are the terminal growth rate. Every other input is held at the values of the previous lesson.
| Required return | Terminal growth 1.5% | Terminal growth 2.0% | Terminal growth 2.5% |
|---|---|---|---|
| 9% | 122.40 | 128.99 | 136.60 |
| 10% | 107.96 | 112.87 | 118.43 |
| 11% | 96.56 | 100.33 | 104.54 |
Dollars a share. The lowest cell is 96.56 and the highest is 136.60, which is a spread of 40.04 dollars, or 41.5 percent of the lower figure.
What the spread is made of
Two assumptions, each moved by one percentage point and half a percentage point respectively. Neither movement is large and neither is unreasonable. A reader who thinks 9 percent is the right required return and one who thinks 11 percent is are not disagreeing about the company at all.
That is worth sitting with. The difference between the highest and lowest cell is larger than the effect of almost anything a reader could learn about the business from further research, and it comes entirely from two numbers neither of which is in any filing.
Reading the grid rather than the point
Three readings are available and only the third is useful.
The point reading takes the middle cell and treats 112.87 as the answer. That discards everything the grid was built to show.
The favourable reading takes the cell nearest whatever conclusion the reader already held. The statistics course has a name for the general form of this and its advice is the same here: a number chosen after seeing the options was not computed, it was selected.
The honest reading takes the whole range as the output, and then asks a separate question: is the price far enough from every cell in the grid for the answer not to depend on which cell is right.
What to widen the grid with
The two inputs above are the two that move a discounted cash flow most, which is why they are the standard grid. At a cyclical business a third belongs beside them: the starting figure.
The first lesson of this course listed three defensible starting figures from the same nine filings, spanning 351 to 1,026 million dollars in individual years around a mean of 883.3 million. Rebuilding the grid on a different starting figure moves every cell proportionally, and a reader working on a cyclical company should do it.
Check yourself
What is the output of the model?
- The middle cell, 112.87 dollars a share
- The whole grid, because the range is what the assumptions produce
- The highest cell, as an upper bound
Show the answer
Answer: B.
The spread from 96.56 to 136.60 comes from moving two assumptions by small, reasonable amounts, neither of which is in any filing.A price sits inside the grid, above four cells and below five. What has the model settled?
- That the shares are fairly valued
- Nothing, because the answer depends entirely on which cell is right
- That the required return should be lowered
Show the answer
Answer: B.
Reporting a conclusion in that case reports the reader's assumptions rather than a finding, and it is the more common of the two cases.What third input belongs in the grid at a cyclical business?
- The share count
- The starting free cash flow figure, because several are defensible from the same filings
- The risk-free rate
Show the answer
Answer: B.
Individual years in this window ran from 351 to 1,026 million dollars around a mean of 883.3. Rebuilding on a different starting figure moves every cell proportionally.Sources
- Toll Brothers, Inc., annual reports on Form 10-K for fiscal years 2017 through 2025, for operating cash flow, capital expenditure and diluted share count; fiscal 2025 accession 0000794170-25-000112, filed 19 December 2025.
What this leaves out
- Every cell in the grid rests on the same starting figure, the same explicit growth rate and the same five-year horizon. Changing any of those moves all nine cells and the grid does not show that.
- The required return and terminal growth values were chosen to span a reasonable range for a worked example. They are not recommendations and no reasoning in this course derives them.
- The two prices in the worked example are illustrative figures chosen to show the two cases. No company's price is described by them.
- Nothing here is a view on Toll Brothers or on any security.