EASTERBROOK, Circuit Judge.

Between 1987 and 1990 a subsidiary of The Sherwin-Williams Company owned 100 percent of the stock of Lyons Group, a motor carrier. Lyons was a losing proposition for Sherwin-Williams (as we call the firm and its subsidiaries, collectively), which not only lost money on operations but also sold the stock for less than it paid. The Central States Pension Fund, a multi-employer fund to which Lyons contributed, believes that the price Sherwin-Williams must pay for this adventure exceeds the business losses: when it sold Lyons, the Fund contends, the entire Sherwin-Williams family of companies withdrew and incurred substantial liabilities under the Multiemployer Pension Plan Amendments Act of 1980, which requires firms that pull out of underfunded plans to chip in. See Milwaukee Brewery Workers' Pension Plan v. Jos. Schlitz Brewing Co., ___ U.S. ___, 115 S. Ct. 981 (1995); Connolly v. PBGC, 475 U.S. 211 (1986); PBGC v. R.A. Gray Co., 467 U.S. 717 (1984); Artistic Carton Co. v. Paper Industry Union-Management Pension Fund, 971 F.2d 1346 (7th Cir. 1992).

Sherwin-Williams has another subsidiary (Dupli-Color) that still contributes to the Fund, but the Fund insists that this does not matter. The argument goes like this. Sherwin-Williams and all of its subsidiaries and affiliates are a single employer by virtue of 29 U.S.C. sec. 1301(b)(1). When it sold Lyons, Sherwin-Williams changed the constituents of its corporate group. The old group, minus Lyons, became a new group. An employer withdraws completely when it "permanently ceased all covered operations under the plan." 29 U.S.C. § 1383(a)(2). Old Sherwin-Williams ceased to exist, just as if it had liquidated and distributed its assets, and thus "permanently ceased all covered operations under the plan." New Sherwin-Williams continued to be a contributing employer, but the fact that one employer joins a plan as another employer withdraws does not relieve the withdrawing employer of its liability. One provision of the MPPAA, 29 U.S.C. § 1398(1)(A), provides that the successor to an employer that winks out of existence because of a "change in corporate structure described in section 1369(b) of this title" is "considered the original employer." Section 1369(b)(3) deals only with corporate "division," and although "division" is an undefined term the sale of a subsidiary's stock is a form of corporate "division." See PBGC Opinion Letters 82-4 (Feb. 10, 1982), 84-7 (Dec. 20, 1984). But sec. 1398(1) applies only if "the change causes no interruption in employer contributions or obligations to contribute under the plan". Six months after its spinoff, Lyons ended operations and filed for bankruptcy. Its demise scotched any exemption under 29 U.S.C. § 1384, which applies only if a business provides security for five years' worth of contributions after the sale. If, as the Fund believes, Lyons' failure was "caused" by the sale, the new Sherwin-Williams corporate group also loses sec. 1398's shelter and must pay approximately $2 million in withdrawal liability. The difference between Lyons' own withdrawal liability of some $1.76 million and the $2 million figure for the entire Sherwin-Williams group likely is less important to the Fund than is the fact that its group-withdrawal approach ensures that the solvent Sherwin-Williams picks up the entire tab (which exceeds the price it realized from the sale of the Lyons stock) whether or not the transaction was designed to "evade or avoid" withdrawal liability, which under 29 U.S.C. § 1392(c) would require Sherwin-Williams to make good Lyons' debt. To collect under sec. 1392(c) a pension trust must establish that avoiding withdrawal liability was a principal purpose of the transaction, and the difficulty of proving intent may explain why the Fund has not pursued this possibility. See Santa Fe Pacific Corp. v. Central States Pension Fund, 22 F.3d 725 (7th Cir. 1994).

Sherwin-Williams and the Fund took their differences to an arbitrator, as the MPPAA requires. The arbitrator stated that the Fund had not presented any claim that the sale of Lyons' stock was designed to "evade or avoid" withdrawal liability or that it represented a "partial withdrawal" of the Sherwin-Williams group under sec. 1385(a). The only question on the table, the arbitrator thought, was whether the sale produced a complete withdrawal of the old Sherwin-Williams group. Observing that Dupli-Color continues to participate in the Fund, the arbitrator concluded that the group's withdrawal is not "complete." As the arbitrator saw things, the Fund is trying to have things both ways: it treats the sale of Lyons' stock as dissolving the old Sherwin-Williams group and substituting a new group, yet it seeks to hold the new group responsible for the old group's liability. If Lyons' departure really caused the old Sherwin-Williams group to withdraw, then the old group logically should be liable; yet looking at the old group's operations would make it plain that withdrawal had not been complete — for in mid-1990 both Lyons and Dupli-Color were active contributors. The Fund sought to collect from the Sherwin-Williams group existing today, which under the Fund's theory is a different employer from the one responsible for Lyons. Moreover, Lyons' failure without surviving for five years meant that it never really became a stand-alone entity, and sec. 1384 attributed its failure to Sherwin-Williams. With Lyons treated as if reabsorbed into the group, neither a change of employer nor a complete withdrawal occurred.

The district judge, entitled to review the arbitrator's legal conclusions independently, see Trustees of Iron Workers Local 473 Pension Trust v. Allied Products Corp., 872 F.2d 208, 211-12 (7th Cir. 1989), took a different route to the same conclusion. 879 F. Supp. 867 (N.D. Ill. 1995). The judge disagreed with the arbitrator's conclusion that Lyons' failure meant that the sale of stock should be disregarded for the purpose of identifying the employer. Only a transaction condemned under sec. 1392(c) should be treated as if it had not occurred, the judge believed, and the Fund has not argued that Sherwin-Williams sold the Lyons stock to "evade or avoid" withdrawal liability. But the judge believed that sec. 1398, on which the Fund does rely, has nothing to do with this case. The statute creating liability for complete withdrawal is sec. 1383(a), which asks whether the employer has ceased making contributions. Only the old Sherwin-Williams group could have ceased contributing (for, on the Fund's theory, the new group did not come into existence until the old group sold Lyons). Yet the old group didn't cut off all contributions; Dupli-Color remains a participating employer. Any attempt to reason backwards from sec. 1398 to the definition of a complete withdrawal, the judge thought, "would really be non-statutory in nature, calling for the judicial recognition of an implied private cause of action where Congress has not expressly created one. And on that score the Supreme Court's increasing reluctance to recognize such implied rights of action should be controlling here." 879 F. Supp. at 877 (emphasis in original).

Although implied rights of action are not in vogue, ERISA and the MPPAA have plenty of express rights of action. In particular, 29 U.S.C. § 1401(b)(2) permits a multi-employer pension fund to recover withdrawal liability. Ours is not a case like Mertens v. Hewitt Associates, 508 U.S. 248, 113 S. Ct. 2063 (1993), or Reich v. Continental Casualty Co., 33 F.3d 754 (7th Cir. 1994), where the provision of one right of action implied the withholding of another. Neither the arbitrator nor the district court directly confronted the Fund's central argument: that when Sherwin-Williams sold its stock in Lyons, the old Sherwin-Williams group vanished, as if it had fallen into one of Kurt Vonnegut's chrono-synclastic infundibulums. Ex-groups necessarily have withdrawn, fully and permanently, and owe withdrawal liability unless some statute prevents collection. Section 1398, the only candidate, is not up to the task, according to the Fund, because the stock sale "caused" Lyons to fail.

Now we have serious doubts that a sale of stock can be blamed for a business failure. Transfer of the certificates' ownership does not affect the firm; that the identity of the investors does not alter business operations is a central feature of corporate law. See Baxter Healthcare Corp. v. O.R. Concepts, Inc., 69 F.3d 785 (7th Cir. 1995). Stock certificates are corporate liabilities — and contingent ones at that, for the owners are not entitled to dividends in the same way debt investors are entitled to interest. If the sale of stock were accompanied by some other event — the sale of assets for an inadequate price, the replacement of top managers, or the assumption of liabilities creating a negative cash flow — then the transaction might alter the firm's prospects. But the Fund does not contend that Sherwin-Williams stripped Lyons of its profitable assets and sold only chaff; it seems to have sold the business intact. If so, then competition from other motor carriers, rather than the sale of stock, must be the prime candidate for an explanation of Lyons' failure; recall that Lyons had been headed downhill while Sherwin-Williams was its sole owner. One could say that Sherwin-Williams was better situated than its successor to invest new capital in Lyons' business, but stockholders are not obliged to throw good money after bad — are not obliged to pony up extra money under any circumstances. That is the meaning of limited liability, and of the rule in most corporate codes that shares are "nonassessable" once paid for. See Model Business Corporation Act sec. 6.21(d) (1984 rev.); see also id. at sec. 6.22. Because the arbitrator ruled against the Fund without addressing the causation question, however, it is not possible to decide the case on this ground. We must assume that the sale of stock caused the collapse. The question remains whether the sale of a subsidiary terminates the existing corporate group and causes a new, shrunken group to spring into existence.

What if Sherwin-Williams had acquired a new subsidiary instead of selling an existing one? If any change in the composition of a corporate group winds up the life of that group, then this acquisition produces a "complete withdrawal" with concomitant liability. So too with the closing of 1 out of 100 subsidiaries that contribute to the Fund; nothing in the Fund's argument logically depends on the fact that Sherwin-Williams had only two contributing subsidiaries. Why stop with the sale or purchase of a subsidiary? What about the opening of a new plant or terminal or corporate division, or the close of an old one? A corporation is just a nexus of contracts, subject to rearrangement in many ways. Section 1301(b)(1) and the implementing regulations are designed to treat subsidiaries like divisions. If selling stock means a complete withdrawal, then so does a change in divisional organization or operation. Cf. Copperweld Corp. v. Independence Tube Corp., 467 U.S. 752 (1984) (treating subsidiaries and divisions alike for purposes of the antitrust laws). And that is not all. What about the sale of a key patent, which may mean more to the firm's operations than any division or subsidiary? Change is the one constant in corporate organization. Employees, plants, products, divisions, subsidiaries, all come and go. Heraclitus quipped that you cannot step twice into the same river. A corporate group is always changing, and the change flows more swiftly than many rivers. "Complete withdrawal" then must be frequent. Many of these will not come within sec. 1398. Return to the 1-in-100 subsidiary example: closing one subsidiary causes an interruption in that corporation's contributions.

Does it make sense of the statute to treat our 1-in-100 example as a "complete withdrawal" of the entire corporate group? The MPPAA draws a sharp line between complete and partial withdrawal. Under sec. 1385 a withdrawal is only "partial" unless the contribution decline exceeds 70 percent, or there is "a partial cessation of the employer's contribution obligation." 29 U.S.C. § 1385(a)(2). If a single corporation has 100 plants, the closing of one will not produce either a full or a partial withdrawal unless the closed plant did 70 percent of the firm's business. (If the closed facility employed more than 20 percent of the firm's covered workers, the substantial-employer rules of 29 U.S.C. § 1363-65 govern. See 29 U.S.C. § 1362(e).) The PBGC believes that closing a subsidiary should be treated just like closing a plant, and that selling all of a subsidiary's assets should be treated just like closing a subsidiary. Opinion Letter 92-1 (Mar. 30, 1992). Yet under the Fund's view, each of these transactions would extinguish the group and create complete withdrawal, because each causes a subsidiary to disappear. Given the statutory objective of treating a firm that organizes into a holding company with 100 subsidiaries just like a corporation organized into 100 divisions, it must follow that a group's sale of a single subsidiary's stock should be treated like selling its assets, or like selling an unincorporated division, and therefore does not produce a complete withdrawal of the group — not unless that subsidiary was the only one making contributions to a particular plan.

Section 1301(b)(1) reinforces this conclusion, albeit indirectly. This section reads: "For purposes of this subchapter, under regulations prescribed by the corporation, all employees of trades or businesses (whether or not incorporated) which are under common control shall be treated as employed by a single employer and all such trades and businesses as a single employer. The regulations prescribed under the preceding sentence shall be consistent and coextensive with regulations prescribed for similar purposes by the Secretary of the Treasury under section 414(c) of Title 26." The PBGC's regulations, in 29 C.F.R. Part 2612, do nothing but point to the IRS regulations under sec. 414. Regulations under sec. 414 do little beside point elsewhere. See 26 C.F.R. sec. 1.414(b)-1. The end of this chain is 26 C.F.R. sec. 1.1563-1, which defines a "controlled group." Section 1.1563-1 is founded on the premise that the departure (or arrival) of a subsidiary does not terminate the controlled group and create a "new group." Instead the regulations speak of "component members" arriving and departing, and they provide rules for determining the effective date of these transactions. 26 C.F.R. sec. 1.1563-1(b). The regulations treat the "controlled group" as having an independent existence, which survives the arrivals and departures that are so numerous in a corporate family of any size. We do not see how the Fund's position can be reconciled with this understanding.

What then is one to make of sec. 1398? As we read it, sec. 1398 is a savings clause, a proviso that comes into play only if some other step causes the participant to vanish. Consider some common reorganizations: (i) A dissolves and distributes its assets to B and C, which carry on the same businesses as A did. (ii) D merges into E, which is the surviving corporation. (iii) F recapitalizes by selling its assets to G, which pays with money raised from issuing debentures. F distributes the cash to its old investors and dissolves. G next merges into H, the source of the capital, which owns all of G's stock. H renames itself " F Corporation." In all three of these cases, and many similar ones, the original participant in the pension plan vanishes under standard doctrines of corporate law, but the business continues. Without some rule allowing the substitution of its successor, a transaction that has no economic effect on the pension fund would generate a windfall of withdrawal liability; and the prospect of this liability could prevent reorganizations that have benefits for investors, employees, and customers alike. Sales of existing subsidiaries, and other transactions that under corporate law leave the parent unaffected, need no such shelter. Both corporate law and the Treasury regulations to which sec. 1301(b)(1) send us treat controlled groups as surviving purchases and sales of stock. The PBGC's Opinion Letter 92-1, to which we have already referred, appears to make a contrary assumption about the effect of sec. 1398, distinguishing the sale of a subsidiary's stock from the sale of a subsidiary's assets, but the opinion letter does not say why. Unexplained assumptions are a poor basis for statutory interpretation.

It may be that some members of Congress, talking on the floor, understood sec. 1398 as the Fund does (although they certainly did not parse its language). Congressional assumptions are some distance from rules of law. American Hospital Ass'n v. NLRB, 499 U.S. 606, 615-17 (1991). The statute delegates to the PBGC (and indirectly to the Secretary of the Treasury) the power to define groups, and that regulatory perspective must be respected. There was, and is, only one "group": the Sherwin-Williams group. No "old Sherwin-Williams group" and no "new Sherwin-Williams group." Just one group comprising changing assortments of assets and operations. That group has not completely withdrawn from the Fund.

One final issue requires only brief treatment. Sherwin-Williams sought attorneys' fees and filed a cross-appeal from the district court's decision requiring each side to bear its own costs and fees both in court, the province of 29 U.S.C. § 1451(e), and before the arbitrator, to which 29 U.S.C. § 1401(a)(2) applies. The right question on appeal is whether the district court abused its discretion. Hooper v. Demco, 37 F.3d 287, 291 (7th Cir. 1994); Nichol v. Pullman Standard, Inc., 889 F.2d 115, 121 (7th Cir. 1989). Cf. Pierce v. Underwood, 487 U.S. 552 (1988), holding that deferential appellate review is the right standard under another statute similar to the attorneys' fees provisions of ERISA. Regulations say that the court should not shift arbitral attorneys' fees unless the loser acts "in bad faith or engages in dilatory, harassing, or other improper conduct during the course of the arbitration". 29 C.F.R. sec. 2641.9(c). This record does not contain a whiff of such conduct. As for the proceedings in the district court: the loser must pay if its position lacked a substantial basis. See Central States Pension Fund v. Lady Baltimore Foods, Inc., 960 F.2d 1339, 1347 (7th Cir. 1992). Although we have established a presumption that one who challenges an arbitrator's award and loses lacked substantial justification, see Continental Can Co. v. Chicago Truck Drivers Pension Fund, 921 F.2d 126 (7th Cir. 1990), a presumption can be overcome. The district court concluded that it had been, because this is a case of first impression with decent arguments on each side. That assessment strikes us as sensible; certainly it is not an abuse of discretion.

AFFIRMED