Textbooks give you a formula for the cost of equity. Filings give you the number somebody was willing to sign.
I went looking for the second kind. The Schedule 14D-9 filed by Potbelly Corporation on 23 September 2025 contains the fairness opinion work of Piper Sandler, and its discounted cash flow section names the rate outright.
Here is the sentence. Piper Sandler "performed discounted cash flow analyses by calculating the range of net present values for each period from second half of fiscal 2025, consisting of the third and fourth fiscal quarters 2025, to fiscal year 2029 based on a discount rate ranging from 12.8% to 17.8%, with a mid-point of 15.3%, reflecting estimates of the Company's weighted average cost of capital."
Five percentage points wide. The mid-point sits at 15.3% exactly.
Stop at the width for a moment. A range that broad is an admission, and it is an honest one. Nobody knows the cost of capital of a mid-cap restaurant business to a decimal place, and a five-point band says so without saying so.
The filing names the method rather than hiding behind a conclusion. Piper Sandler "derived these discount rates by application of the capital asset pricing model, which requires certain Company-specific inputs, including a market capitalization size risk premium, as well as certain financial metrics for the United States financial markets generally."
That gives three ingredients, then. A market model, a company-specific input, and a size premium. The filing calls the first one "the capital asset pricing model" and leaves the arithmetic unpublished.
That last one is the piece I think most people skip. A size risk premium exists because small listed companies have historically earned more than the plain model predicts, and the practical effect is to push the rate up for exactly the kind of company most likely to be sold.
It is not there to be interesting. It is there to produce a range that the board can compare against the price on the table.
The price was "$17.12 per Share in cash". The comparison the filing sets out, on the precedent transactions method, is blunt: a merger consideration implied multiple of "19.9x" against a reference range of "5.3x–20.9x", and an implied equity value per share reference range of "$5.17–$17.96".
Look at that band. It runs from five dollars to eighteen. I had to read the line three times to be sure I had not misread a decimal point.
A range that wide will contain almost any price you care to name, and the deal price sits near its top. I read that as the method being weak for this business rather than the price being generous, though I hold that loosely.
Both inputs matter and only one of them gets discussed.
The same analysis used "EBITDA exit multiples of 10.3x to 12.3x, with a mid-point of 11.3x, based on the median of the selected precedent transactions analysis." That terminal assumption usually drives more of a DCF output than the discount rate does, particularly over a forecast this short.
The forecast window runs from the second half of fiscal 2025 to fiscal year 2029. Four and a half years of projections, and then a multiple decides most of the answer. The exit assumption rests on a median of past deals, which is a choice and not a fact.
I used to argue that the discount rate was the pressure point in a fairness opinion. Wrong, or at least badly incomplete.
The pressure point is the pair. A 15.3% mid-point with an 11.3x exit multiple is a different valuation from the same rate with a 9x multiple, and the filing gives both numbers precisely because either alone tells you nothing. I had been treating the rate as the argument and the multiple as housekeeping.
Nothing to do with valuation. Skip it if you want.
Buried in the appraisal-rights discussion is a different rate entirely. Interest on a court-determined fair value accrues "at 5% over the Federal Reserve discount rate (including any surcharge) as established from time to time", compounded quarterly.
The statute adds that the interest "will be compounded quarterly", with no band and no judgement. That is the rate a dissenting shareholder earns for being right. It has nothing to do with the cost of equity and I keep thinking about the contrast anyway: one rate estimated in a five-point band by a bank, the other fixed by statute to the basis point.
That was the aside. Back to the cost of capital.
Start with the inputs themselves. The filing names the model and the size premium but does not publish the beta, the risk-free rate, or the equity risk premium used, so I cannot reconstruct 15.3% from the ground up. Nobody outside the engagement can do it. The filing does concede that "No transaction utilized in the selected precedent transactions analysis is identical to the Company", which is the closest thing to a warning label here. My guess is that the beta came from a small peer set, and that is a guess.
I also do not know how the five-point width was chosen. A band that wide is a judgement about uncertainty, and no published sentence explains the judgement.
And I am not going to pretend 12.8% to 17.8% is a benchmark for anything. It is one restaurant business, in one market, in one autumn. I had assumed a rate like this was reusable across a sector, and it is not.
If you are reading a fairness opinion, find three numbers before you read a word of the prose: the discount rate range, the exit multiple range, and the width of the resulting per-share band.
In this file those were 12.8%–17.8%, 10.3x–12.3x, and $5.17–$17.96 against a $17.12 price. The width is the tell.
What still bothers me is that the discount rate is the number everyone argues about, and it is the one input in that list the bank explains least.