
675 F.3d 646 (2012)
Tariq MALIK, Mahdiur Rahman, and Janice Quinn, Personal Representative of the Estate of Joe Lee Lott, Plaintiffs-Appellants,
v.
FALCON HOLDINGS, LLC, and Aslam Khan, Defendants-Appellees.
No. 11-2815.
United States Court of Appeals, Seventh Circuit.
Argued February 21, 2012.
Decided March 14, 2012.
*647 Mary Massaron Ross (argued), Attorney, Plunkett & Cooney, Detroit, MI, for Plaintiffs-Appellants.
Samuel G. Harrod, IV (argued), Attorney, Meltzer, Purtill & Stelle LLC, Schaumburg, IL, for Defendants-Appellees.
Before EASTERBROOK, Chief Judge, BAUER, Circuit Judge, and SHADID, District Judge.[*]
EASTERBROOK, Chief Judge.
Falcon Holdings was organized in 1999 to own and operate 100 fast-food restaurants. Aslam Khan owned 40% of Falcon's common units. (Falcon is a limited liability company rather than a corporation; ownership is represented by units rather than shares.) The remainder of the common units, and all of the preferred units, were owned by Sentinel Capital Partners II and Omega Partners (collectively "Sentinel"). According to the plaintiffs, Khan told Falcon's managers that he would acquire full ownership one day, and that, when he did, he would reward the top managers with 50% of Falcon's equity. Plaintiffs say that they accepted lower salaries because they anticipated receiving a stake if Falcon proved to be a success, and that they worked hard to make it prosper (which it did).
Sentinel was bought out in 2005, and Khan became Falcon's sole equity owner. He did not distribute common units to any of the top managers and has denied ever promising that he would. Five of the managers filed this suit. The district court assumed that the evidence in the summary-judgment record would permit a jury to conclude that Khan had promised the plaintiffs an equity stake in Falcon. (Contracts for the sale of stock are not subject to the statute of frauds in Illinois, see 810 ILCS 5/8-113, so the absence of a writing signed by Khan is not dispositive.) Two of the original plaintiffs nonetheless lost on the basis of releases; they have not appealed. The others lost because, the district judge held, they had not adequately estimated the damages they sustained. 2011 WL 2790168, 2011 U.S. Dist. LEXIS 77983 (N.D.Ill. July 15, 2011). These three have appealed. (One has died; his estate's representative has been substituted.)
Plaintiffs offer a simple estimate of damages. They calculate that the price paid for Sentinel's ownership interest (100% of the preferred units and 60% of the common units) implies that Falcon as a whole was worth approximately $48 million in 2005. Half of $48 million is $24 million. In 2005, twenty managers qualified for units under the terms of Khan's offer. Thus each plaintiff lost about $1.2 million when Khan did not keep his promise.
The district court stated that plaintiffs' approach has two flaws, each fatal: first, because Sentinel did not own 100% of Falcon, it is impossible to derive the value of the whole firm from the amount paid for its holdings; second, the amount that Sentinel was paid depended on how much *648 Khan and Falcon could borrow rather than Falcon's true value. Neither of these propositions is sound; indeed, each supposes that there is some measure of "true" value that differs from what a willing buyer will pay a willing seller in an arms'length transaction. Yet that is the gold standard of valuation; other measures are approximations. The value of a thing is what people will pay. The judiciary should not reject actual transactions prices when they are available.
Let us simplify the transaction by assuming that Sentinel owned 60% of Falcon and accepted $6 million for its units. Falcon as a whole then must be worth at least $10 million. If it is worth less than that, Khan has overpaid. Khan does not contend in this litigation that he paid Sentinel too much. Falcon might be worth more than $10 million in this example; Sentinel would accept "only" $6 million if it thought that Khan, as the controlling manager, would prevent Sentinel from receiving payments equivalent to 60% of the firm's full value. But if the price Sentinel accepted represents less than 60% of Falcon's value, then plaintiffs have underestimated their damages. A court can't dismiss a suit because the plaintiffs are asking for less than their due.
The same thing is true about the district court's belief that the ability of Khan and Falcon to borrow money set a cap on what Sentinel received. If this means that Sentinel accepted less for its units than their proportional share in Falcon represented, then again plaintiffs have underestimated their damages. That's not a good reason why they should go home empty-handed.
There's another problem with this aspect of the district court's analysis. The amount that Khan and Falcon could borrow depended on Falcon's value. Although the record surprisingly does not contain the details of the transaction, it appears to be a leveraged buyout (LBO). In an LBO, a business borrows money against its own value, promising to repay from its anticipated net earnings. Outside investors are cashed out; insiders own the equity in a highly leveraged venture. The amount a firm can borrow to conduct an LBO depends on the lender's estimate of its future earnings, which is a good indicator of value. So to say that Falcon could not pay Sentinel more than Falcon could borrow is not to say that the price was an arbitrary number. If the amount offered were a poor estimate of Falcon's value, Sentinel would have said no. Instead it took the offer. To repeat, we have a willing buyer and willing seller dealing at arms' length; the price they agree on is the value of the asset.
The real problem with plaintiffs' damages estimate is not inability to value Falcon Holdings as an entity. It is that what Khan promised was half of the equity interest in Falcon. Khan emerged from the LBO owning 100% of the equitybut not 100% of Falcon. Suppose Falcon borrowed $38 million from a bank (or syndicate of banks) to pay off Sentinel. Then, if Falcon was worth $48 million as a whole in 2005, the lenders' debt interest was $38 million and Khan's equity interest was worth $10 million. Half of that, split 20 ways, would come to $250,000 for each plaintiff, not the $1.2 million apiece they have demanded. Because the record does not contain the details of the transaction, we have no idea whether this example is even approximately accurate. But it is unsound to assume, as plaintiffs do, that Khan's equity interest in Falcon is worth 100% of the firm's value. It might take an expert financial economist to derive an equity valuation, and plaintiffs did not disclose an expert in discovery. For their part, however, defendants have not asked *649 us to affirm on the ground that the record is silent about the value of Khan's equity interest in Falcon.
In addition to assuming that the value of Falcon as a whole is the same as the value of Khan's equity interest, plaintiffs make a second questionable assumption: that Khan would hand over to each of the 20 managers 2.5% of Falcon's equity units without any terms or conditions. That would be a disaster not only for the ownership structure of a closely held firm but also from a tax perspective. The units' value would be taxed as ordinary income, just like a cash bonus. To pay the tax, many of the managers might have had to sell some or all of their unitsyet there is no market for units in a limited liability company. To avoid problems such as these, firms usually distribute options rather than shares (or units). The exercise price of the options will be set at the value of the shares (or units) on the date the options are awarded, so there is no taxable income until the options are exercisedand then the tax is at the capital gains rate rather than the higher ordinary-income rate. Options not only have tax benefits but also offer managers a share in any appreciation without the risk of capital loss. (Rewards for past success can be distributed as bonuses, also without exposing managers to loss from future operations.) Many managers hold under-diversified portfolios and are risk averse as a consequence. Exposing them to a risk of capital loss could injure the firm by inducing them to be timid when making decisions.
Perhaps Khan indeed offered Falcon's managers illiquid units that would require them to pay immediate taxes without a means to raise the money to do so, and without any terms such as buy-sell agreements (which would provide a means for valuing the units and preventing distribution to outsiders). But such a transaction would be sufficiently unusual that plaintiffs cannot simply assume that a promise to give them an equity interest in the firm was to be accomplished by handing out units rather than options. If ownership would have entailed options, then it becomes necessary to know the exercise price, the duration of the options, and at what rate they would vest. (Deferred vesting is common in order to give managers an incentive to remain with the firm.) So many vital terms are missing that any promise may well be too indefinite to enforce, see Brines v. XTRA Corp., 304 F.3d 699 (7th Cir.2002); ATA Airlines, Inc. v. Federal Express Corp., 665 F.3d 882 (7th Cir.2011)but once again defendants have not asked us to affirm on this ground.
Defendants do try to defend their judgment by arguing that plaintiffs waited too long to quantify their damages. According to defendants, details should have been set out before the close of discovery, perhaps as early as the initial disclosures under Fed.R.Civ.P. 26(a)(1)(A), and plaintiffs' delay entitles them to prevail outright. This is absurd. Litigants are entitled to use discovery to learn facts (such as how much Sentinel received in the buyout) that will affect the remedy; a party can wait until the facts are in hand before adding specifics to the claim adumbrated in the complaint. Anyway, if defendants thought that plaintiffs had failed to perform their obligations under the rules, they should have asked the district judge for a sanction before discovery closed rather that waiting (as they did) until their motion for summary judgment. Fed.R.Civ.P. 37(c)(1) gives the judge discretion to match a remedy to the wrong. Defendants have not explained how the plaintiffs' delay injured them, so a remedy (if any rule was transgressed) would have been mild. (We have explained in Ball v. *650 Chicago, 2 F.3d 752 (7th Cir.1993), and many other cases, that the remedy for procedural missteps in litigation must be proportionate to the injury.) As defendants did not ask for any arguably appropriate sanction, however, they are in no position to complain that the district judge did not award one.
The judgment is vacated, and the case is remanded for proceedings consistent with this opinion.
NOTES
[*]  Of the Central District of Illinois, sitting by designation.
