In this action for a refund of federal corporate income taxes, the district court granted summary judgment to the government. We reverse.
I.
This case involves three separate-but-related corporate entities — Proctor & Gamble (“P & G”), Proctor & Gamble of Canada (“P & G-Canada”), and Proctor & Gamble Foreign Sales Corporation (“P & G-FSC”) — each of which files its own separate tax return with the Internal Revenue Service (IRS). P & G is a producer of consumer products — a Cincinnati, Ohio-based corporation that produces goods such as Crest toothpaste, Duracell batteries, Ivory soap, and Pringles potato chips, and is the parent company of the other two. P & G-Canada is a distributor — a wholly-owned subsidiary that sells P & G’s products in Canada. P & G-FSC is a tax-shelter — a wholly-owned subsidiary (though actually just a shell corporation) incorporated off-shore (Barbados) for the sole purpose of reducing P & G’s federal income tax on its exports, pursuant to a former provision in the tax code (i.e., the “FSC Program,” 26 U.S.C. §§ 921-927, since repealed).1
In the tax years ending in 2000 and 2001, P & G produced goods that P & G-Canada sold to Canadian consumers, and P & G-Canada paid for those goods. But, for tax purposes, P & G-Canada actually paid P & G-FSC, which in turn paid P & G. Although central to the present case, this transaction (which the parties refer to as the “Advance Payment Transaction”) was only a small part of the companies’ business in the 2000 and 2001 tax years and one small component of their tax returns for those years. The “advance payment” aspect of this transaction resulted in P & G-FSC’s and P & G’s recording their income from the transaction in tax year 2000 and their expenses for the transaction in 2001.2 During discovery, P & G and P *470& G-FSC reported, via interrogatories, that they had, in their tax filings, attributed the following incomes [I] and expenses [E] to this transaction:
Company Year Item Amount Tax Return
P & G-PSC 2000 Sales [I] $374,790,000 Form 1120FSC, Sch. B, Line 1
2000 Costs (other) [E] $ 2,721,689 Form 1120FSC, Sch. G, Line 14
2001 Loss Reimbursement (from P&G) [I] $288,588,299 Form 1120FSC
2001 COGS 3 [E] $288,588,300 Form 1120FSC, Sch. P, Pt. Ill, Lines 22 and 23
P&G 2000 Sales [I] $288,588,300 Form 1120, Line 1
2001 COGS [E] $359,344,974 Form 1120, Line 2
2001 Loss Reimbursement (to P & G-FSC) [E] $288,588,299 Form 1120, Line 26
Neither the IRS nor the district court disputed the accuracy or validity of any of this information, and we will therefore accept this information for purposes of our later calculations and analysis.
Following completion of the 2000 and 2001 tax years, P&G calculated its4 corporate income taxes for those years (applying its own interpretation of the FSC Program provisions), filed its tax returns with the IRS, and paid taxes in the amounts calculated. The IRS audited the returns, made several adjustments, and ordered P&G (and P & G-FSC) to pay additional taxes. P&G and P & G-FSC paid the additional taxes as ordered; filed a claim for a refund with the IRS pursuant to 26 U.S.C. § 6402 (i.e., exhausted administrative challenges to the imposition of those additional taxes), which was denied; and filed a complaint against the IRS in federal court pursuant to 26 U.S.C. § 7422.
In their complaint, P&G and P & G-FSC challenged several of the IRS’s adjustments and settled all but one, the Advance Payment Transaction. The parties filed cross-motions for summary judgment and the court began its opinion by describing the transaction, explaining that P & G-Canada had paid P & G-FSC which had in turn paid P&G, and that the “buyers” (P & G-Canada and P & G-FSC) had paid in advance, allowing the “sellers” (P&G and P & G-FSC) to record income in tax year 2000 but expenses in tax year 2001. The district court framed the question in terms of this separation between income on the 2000 tax returns and expenses on the 2001 returns: whether this was allowable under the FSC Program’s Combined Taxable Income (CTI) provision, 26 U.S.C. § 925(a)(2) and 26 C.F.R. § 1.925(a)-IT(c)(6), or whether the CTI provision required the taxpayer to “match” expenses (costs) with income (payments) in completing the CTI calculation. Put another way, the question was whether the CTI calculation is transaction specific. Or, to be a little more explicit, whether the taxpayer must include all income and all expenses attributable to any given transaction, for any and every transaction that the taxpayer includes in the CTI rubric during a given tax year, regardless of the year in which the income or expense actually occurred, regardless of prevailing accounting practices, and regardless of other statutory or regulatory requirements.
The district court ruled in favor of the IRS, holding that, for any given transac*471tion, all expenses and all income attributable to that transaction must be included in the CTI calculation. The district court cited General Dynamics Corp. v. C.I.R., 108 T.C. 107 (U.S. Tax Court 1997), for the proposition that the CTI provision supersedes prevailing accounting practices and other regulatory requirements (such as accrual-based accounting) to require a matching of all incomes and expenses. The district court then cited Boeing Company v. United States, 537 U.S. 437, 123 S.Ct. 1099, 155 L.Ed.2d 17 (2003), for the proposition that the CTI calculation’s result must “approximate an arm’s-length transaction.”
P & G moved the court to reconsider, but the court declined. P & G then moved the court to clarify and modify the summary judgment order, so as to allow P & G to compute its taxes under the “gross receipts” method, 26 U.S.C. § 925(a)(1), rather than merely rejecting its refund claim and binding it to the IRS’s pre-trial approach and calculations. The court invoked the variance doctrine, see Estate of Bird, 534 F.2d 1214, 1219 (6th Cir.1976) (“[T]he grounds on which a claim for a tax refund is made must be specifically set forth in the claim for refund itself, otherwise the court in a refund action is without jurisdiction to consider them.”), and, finding that P & G had not set forth its gross-receipts argument in the refund claim submitted to the IRS, denied the motion.
The district court entered final judgment and P & G and P & G-FSC appealed. Upon consideration of the parties’ arguments, review of the record, and analysis of the pertinent law, we find that we are presented with four sequential questions: (1) did the district court err in its analysis and conclusion; (2) if so, did P & G misapply § 925, (3) if so, did the IRS improperly impose its own FSC method (in violation of its own regulations), and (4) if so, does the variance doctrine apply?
II.
In a tax refund case, we calculate the proper amount of tax due under a correct interpretation of the applicable provisions. See Jones v. Liberty Glass Co., 332 U.S. 524, 531, 68 S.Ct. 229, 92 L.Ed. 142 (1947). In this case we have calculated all of the possibilities (and competing arguments) for comparative purposes.
A. Ordinary Corporate Income Tax Liability (hypothetical)
In the simplest sense, P & G could (hypothetically) calculate its profit from the Advance Payment Transaction by ignoring the FSC benefit and merely subtracting Costs from Sales:
P & G (total transaction — No FSC)
Sales Income $374,790,000
COGS 359,344,974
Costs (other) $ 2,721,689
Taxable Income5 $ 12,723,337
Tax (35%) $ 4,453,168
So, because United States corporations pay 35% tax at this level of income, see 26 U.S.C. § 11(b)(1)(D), in this hypothetical, P & G would owe the IRS $4.45m in corporate income tax.
But, in order to reduce its taxes, P & G included P & G-FSC in the transaction (reduced taxes being the FSC’s raison d’etre), which complicates the analysis a bit. However, this is simple in concept. Because the IRS taxes FSCs at a lower effective rate, as explained below, P & G could reduce the total tax liability for this transaction by including P & G-FSC in the transaction and sharing the pre-tax profit (i.e., “taxable income”) with P & G-FSC. And, because P & G-FSC is P & G’s *472wholly-owned subsidiary, this “sharing” would not actually diminish P & G’s overall profit. It should also be evident that P & G could most reduce its tax liability (and correspondingly maximize its overall profit) by directing all of its profit to P & G-FSC, at its lower tax rate. Hypothetically:
Without P & G-FSC With 100% toP& G-FSC Difference
Taxable Income: $12,723,337 Taxable Income: $12,723,337
Tax (35%): 4,453,168 Tax (12.17%): 1,548,928 in Tax: $2,904,240
After-Tax Profit: 8,270,169 After-Tax Profit: 11,174,409 in Profit: 2,904,240
So, using the FSC tax rate6 of 12.17%, if P & G could direct all of its profit to P & G-FSC, then its total tax liability would be only $1.55m, and it would benefit by over $2.90m. But this is not allowed — the FSC Program had limitations, one of which was that the company could not simply substitute the FSC in its place; that is, P & G could not direct all of its profit to P & G-FSC.
B. Tax Liability via the FSC Program
The FSC Program offered three ways by which a FSC could participate in the transaction: as a pass-through, see 26 U.S.C. § 925(a)(1); as a partner, see § 925(a)(2); or as an independent middle man, see § 925(a)(3). Consider each of these, as applied to P & G and P & G-FSC.
1. Pass-Through (or Gross Receipts) Method
Pursuant to 26 U.S.C. § 925(a)(1), the taxable income of the parent company and the FSC “shall be based upon a transfer price which would allow [the] FSC to derive taxable income attributable to [the] sale (regardless of the sales price actually charged) in an amount which does not exceed ... 1.83 percent of the foreign trading gross receipts derived from the sale.” So, in this scenario, P & G-Canada would pay P & G-FSC (for the goods received from P & G); P & G-FSC would keep 1.83% of that payment (i.e., the “gross receipts”) as profit (“taxable income”), passing the rest along to P & G; and P & G would calculate its profit (“taxable income”) by subtracting the cost of goods sold (“COGS”) from the money received from P & G-FSC (i.e., “sales income”). P & G and P & G-FSC would calculate their tax liability on their respective profits at their respective rates.
p& G-FSC P&G Total
Sales Income [SI]: $374,790,000 Sales Income: $365,209,654
Costs: 2,721,689 COGS: 359,344,974
Profit [1.83% of SI]: 6,858,657 Profit [SI — COGS]: 5,864,680 Pre-Tax Profit: $12,723,337
Taxes (12.17%): 834,967 Taxes (35%): 2,052,638 Total Tax: 2,887,605
After-Tax Profit: 6,023,690 After-Tax Profit: 3,812,042 Total A-T Profit: 9,835,732
*473Note that the total pre-tax profit is the same as above ($12.72m), but the total tax liability is reduced by $1.57m (from $4.45m to $2.89m, or 35%) and the total after-tax profit is correspondingly increased by $1.57m (from $8.27m to $9.84m, or 19%). And, we should note that P & G’s (and P & G-FSC’s) use of accrual-based accounting, see 26 U.S.C. §§ 451 & 461, in which income is recorded in the first year and costs are recorded in a subsequent year, would not change this outcome.
Fiscal Year 2000
P&G-FSC P&G Total
Sales Income: $374,790,000 Sales Income: $365,209,654
Costs: 2,721,689
Profit [1.83% of SI]: 6,868,657 Profit [SI — COGS]: 365,209,654
Taxes (12.17%): 834,967
After-Tax Profit: 6,023,690
Fiscal Year 2001
P& G-FSC P&G Total
Sales Income: Sales Income: $0
Costs:
Profit [1.83% of SI]: Profit [SI — COGS]: (359,344,974)
Taxes (12.17%): Taxes (35%); (125,770,741)
After-Tax Profit: After-Tax Profit: (233,574,233) 2001 A-T Profit: (233,574,233)
Combined 2000-2001 Total
Profit: $12,723,337
Taxes: 2,887,605
A-T Profit: 9,835,732
The total pre-tax profit is the same as for the foregoing (cost-based accounting) calculation ($12.72m), as is the total tax liability ($2.89m) and the total after-tax profit ($9.84m).
2. Partnership (or CTI) Method
Pursuant to 26 U.S.C. § 925(a)(2), the taxable income of the parent company and the FSC “shall be based upon a transfer price which would allow [the] FSC to derive taxable income attributable to [the] sale (regardless of the sales price actually charged) in an amount which does not exceed ... 23 percent of the combined taxable income of such FSC and such person [i.e., the FSC and the parent company together] which is attributable to the foreign trading gross receipts derived from the sale.” And the IRS defined this “combined taxable income” (“CTI”) as “the excess of the foreign trading gross receipts of the FSC from the sale over the total costs of the FSC and related supplier including the related supplier’s cost of goods sold.” 26 C.F.R. § 1.925(a)-lT(c)(6).
In our scenario, P & G-Canada would pay P&G and P & G-FSC together, as partners; this joint entity (P & G+P & G-FSC) would calculate the total combined profit (i.e., the money received from P & G-Canada minus P & G’s costs and P & G-FSC’s costs); and the two entities would apportion that profit, with 28% to P & G-FSC and 77% to P & G. After apportionment, the two entities would calculate their tax liability separately, on their respective profits at their respective tax rates.
*474 P & G + P & G-FSC combined, entity 7
Sales Income: $374,790,000
P&G Costs (COGS): 359,344,974
P & G-FSC Costs (other): 2,721,689
Combined Taxable Income (CTI): 12,723,337
P & G-FSC P&G Total
Portion of CTI (23%): $2,926,368 Portion of CTI (77%); $9,796,969 Total $12,723,337
Taxes (12.17%): 356,253 Taxes (35%): 3,428,939 Total Taxes: 3,785,193
After-Tax Profit: 2,570,114 After-Tax Profit: 6,368,030 A-T Profit: 8,938,144
Note that the total pre-tax profit is the same as above ($12.72m), but the total tax liability in this scenario is $3.79ra, which is less than the tax liability without the FSC ($4.45m) but more than the tax liability in the pass-through scenario ($2.89m). Correspondingly, the total after-tax profit is more than it was without the FSC ($8.27m) but less than in the pass-through scenario ($9.84m). And, as with the pass-through scenario, note that the use of accrual-based accounting, in which income is recorded in the first year and costs in the next, would not change this outcome.
Fiscal Year 2000: P & G+P & G-FSC combined entity
Sales Income: $374,790,000
P&G Costs (COGS): 0
P & G-FSC Costs (other): 2,721,689
Combined Taxable Income (CTI): 372,068,311
P & G-FSC P&G Total
Portion of CTI (23%): $86,575,712 Portion of CTI (77%): $286,492,599 Total
After-Tax Profit: 75,157,799 After-Tax Profit: 186,220,190 A-T Profit:
Taxes (12.17%): 10,417,913 Taxes (35%): 100,272,410 Total Taxes: 110,690,323
*475Fiscal Year 2001: P & G+P & G-FSC combined entity
Sales Income:
P&G Costs (COGS):
0 P & G-FSC Costs (other):
Combined Taxable Income (CTI): (359,344,974)
P & G-FSC P&G Total
Portion of CTI (23%): ($82,649,344) Portion of CTI (77%): ($276,695,630) Total ($359,344,974)
Taxes (12.17%): ( 10,061,659) Taxes (35%): ( 96,843,471) Total Taxes: ( 106,905,130)
After-Tax Profit: ( 72,587,685) After-Tax Profit: ( 179,852,159) A-T Profit: ( 252,439,844)
Combined 2000-2001 Total
Total 12,723,337
Total Taxes: 3,785,193
A-T Profit: 8,938,144
So the total profit is the same here as for the single calculation (cost-based accounting) above ($12.72m), as is the total tax liability ($S.79m) and the total after-tax profit ($8.94m).8
3. Independent Middle Man (or Arm’s-Length Negotiation) Method
Pursuant to 26 U.S.C. § 925(a)(3), the taxable income of the parent company and the FSC “shall be ... [the] taxable income based upon the sale price actually charged (but subject to the rules provided in [26 U.S.C. § ] 482).”9 So, in this scenario, P *476& G-Canada would pay P & G-FSC an amount determined via “arms-length” negotiation, as though P & G-FSC were an unrelated middle man; P & G-FSC would similarly pay P & G an amount determined via “arms-length” negotiation and keep the difference as profit (“taxable income”); and P & G would calculate its profit (“taxable income”) by subtracting its costs from the amount received from P & G-FSC. P & G and P & G-FSC would calculate their tax liability on their respective profits at their respective rates.
This is the approach the IRS advocated as its “primary” position.10 The “price actually charged” by P & G-FSC to P & G-Canada was known — it was $374,790,000 (the amount P & G-Canada actually paid P & G-FSC for the goods). The unknown was the “price actually charged” by P & G to P & G-FSC, and that is because there had never actually been an arm’s-length negotiation between those two. The IRS proposed a P & G-to-P & G-FSC “arm’s length” price of $374,790,000 — the entire amount P & G-FSC had received from P & G-Canada. The IRS explained that, “[tjypically, the activities of the FSC (functions performed, assets employed, risks assumed, etc.) would not entitle the FSC to earn more than a de minimis amount of income from the transactions.”
So, in the IRS’s view, P & G-FSC was not entitled to any profits because it had not performed any functions, employed any assets, or assumed any risks. But, de minimis is different from none and if the FSC could not share in the profits, then P & G could not reduce its tax liability and would have no reason to involve the FSC at all. For our purposes, let’s assume that P & G sold its goods to P & G-FSC at 2.5% above cost (i.e., at 2.5% profit), which seems reasonably “de minimis”:
P &G-FSC P&G Total
Sales Income: $374,790,000 SI [COGS + Profit]: $368,328,598
Costs (incidental): 2,721,689 Costs (incidental): 0
COGS [P & G’s SI]: $368,328,598 COGS: 359,344,974
Profit: 3,739,713 Profit [2.5% COGS]: $ 8,983,624 Profit: $12,723,337
Taxes (24.5%): 916,230 Taxes (35%): 3,144,269 Total Taxes: 4,060,498
After-Tax Profit: 2,823,483 After-Tax Profit: 5,839,356 Total A-T Profit: 8,662,839
Note that the FSC tax rate11 in this approach is 24.5% rather than the 12.17% available to the other two approaches. See 26 U.S.C. § 291(a)(4)(A). Note also that the outcome is reasonably consistent with the prior two approaches. The total pretax profit is the same as above ($12.72m), the total tax liability is $4.06m, which is less than the tax liability without the FSC ($4.45m) but more than the tax liability in the other two scenarios. Correspondingly, the total after-tax profit is more than it was without the FSC ($8.27m) but less than in the other two scenarios.
C. P & G’s Calculation of Tax Liability
For tax years ending 2000 and 2001, P & G and P & G-FSC calculated their tax liability based on their interpretation of the Combined-Taxable-Income (CTI) method, § 925(a)(2), and the “no loss rule,” 26 C.F.R. § 1.925(a)-IT(e)(l)(i),12 a provi*477sion which allows the parent company to reimburse the FSC for the costs incurred in conducting the transaction. The end result here, facilitated in part by their (otherwise acceptable) use of accrual-based accounting, was an enormous tax benefit:
Fiscal Year 2000
P&G-FSC P&G Total
Sales Income: Income:
Costs: 0 Costs (COGS):
Profit [SI — Costs]: 374,790,000 Profit [SI — Costs]: 288,588,300 FY2000
Taxes (12.17%): 45,626,609 Taxes (35%): 101,005,905 FY2000 Taxes: 146,632,514
After-Tax Profit: 329,163,391 After-Tax Profit: 187,582,395 2000
Fiscal Year 2001
P&G-FSC P&G Total
Income: 13 $288,588,300 Sales Income:
Costs (COGS): 288,588,300 Costs (COGS): 359,344,974
Costs (other) 0 Costs (other): 288,588,300
Profit [I — Costs]: 0 Profit [SI — Costs]:
Taxes (12.17%): 0 Taxes (35%):
After-Tax Profit: 0 After-Tax Profit: (421,156,628) 2001 A-T Profit: ( 421,156,628)
Combined 2000-2001
P& G-FSC P&G Total
Pre-Tax Profit: 374,790,000 Pre-Tax Profit: (359,344,974)
Taxes: 45,626,609 Taxes:
After-Tax Profit: 329,163,391 After-Tax Profit: (233,574,233) After Tax Profit: 95,589,168
The first, most obvious, thing to notice about this calculation is that the tax liability is now a tax benefit of $80.14m and the total after-tax profit is $95.59m, which is a “substantial” improvement over the original result without the FSC (from $8.27m to $95.59m, or 1,056%).
The dispute central to this case concerns competing interpretations of the CTI method, § 925(a)(2), but we cannot realistically consider or resolve that dispute until we address and resolve two other issues with P & G’s calculations. The first is that this calculation does not capture P & G-FSC’s $2.72m in “other costs.”14 For a comparison of the competing results to be meaningful, the underlying calculations must include identical incomes and expenses. Therefore, we must include this $2.72m expense. The second outstanding issue concerns the $288m that P&G “paid” to P & G-FSC in 2001 as “commission revenue” for “loss reimbursement,” purportedly pursuant to 26 C.F.R. § 1.925(a)-IT(e)(l)(i) (the “no-loss rule”), and which the IRS labeled a “meaningless circular cash flow.”15 According to the *478parties, this issue is currently under dispute elsewhere, see Appellee’s Brief at 14 n. 5, and is not presently before us. But, as with the first issue, for a comparison of the results to be meaningful, we must omit16 this “meaningless circular cash flow” from our present calculations. This results in the following, modified, version of P & G’s tax liability calculations:
Fiscal Year 2000
P & G-FSC P&G Total
Sales Income: $374,790,000 Sales Income:
Costs (COGS): 0 Costs (COGS): 0
Costs (other): 2,721,689 Costs (other): 0
Profit [SI — Costs]: .372,068,311 Profit [SI — Costs]: FY2000 Profit:
101,005,905 FY2000 Taxes: 146,301,178
326,773,038 Profit: 187,582,395 2000 A-T Profit: 514,355,433
Fiscal Year 2001
P& G-FSC P&G Total
Income: Sales Income:
Costs (COGS): 288,588,300 Costs (COGS):
Costs (other) 0 Costs (other): 0
Profit [I — Costs]: (288,588,300) Profit [SI — Costs]: FY2001 Profit:
Taxes (12.17%): (35,132,489) Taxes (35%): FY2001
After-Tax Profit: (253,455,811) After-Tax Profit: (233,574,233) 2001 A-T Profit: (487,030,044)
Combined 2000-2001
P & G-FSC P&G Total
Pre-Tax Profit: After-Tax Profit: 83,480,011 10,162,784 73,317,227 Pre-Tax Profit: After-Tax Profit: (70,756,674) (24,764,836) (45,991,838) Pre-Tax Profit: Taxes: After Tax Profit: $ 12,723,337 (14,602,052)
So, having resolved these two concerns, we find that under P & G’s interpretation and application of the CTI method, § 925(a)(2), the tax liability is still actually a tax benefit, albeit a much smaller one ($14.60m rather than $80.14m), and the total after-tax profit is $27.33m, which remains a “substantial” improvement over the result without the FSC (from $8.27m to $27.33m, or 230%).
D. IRS’s Calculation of Tax Liability
The IRS deemed P & G’s application of the CTI method “impermissible,” finding that it resulted in a “material distortion of income” under the pertinent regulation, which says:
A FSC may, generally, choose any method of accounting permissible under section 446(c)[17] and the regulations under *479that section. However, ... the FSC may not choose a method of accounting which, when applied to transactions between the FSC and [the parent company], will result in a material distortion of the income of the FSC or of [the parent company]. Changes in the method of accounting of a FSC are subject to the requirements of section 446(e)[18] and the regulations under that section. 26 C.F.R. § 1.925(a) — lT(c)(6)(iii)(B). The IRS then held that, because P&G had misinterpreted and/or misapplied the CTI method, the first two FSC Program approaches (i.e., the pass-through and the partnership approaches) were no longer available, and P&G was bound to an independent-middle-man (i.e., § 925(a)(3) arm’s length) approach, as calculated by the IRS. That was:
Fiscal Year 2000
P& G-FSC P&G Total
Sales Income: $374,790,000 Sales Income: $374,790,000
Costs: 0 COGS: 0
Profit: 374,790,000 Profit: 374,790,000 FY2000 Profit: $749,580, 000
Taxes (12.17%): 45,611,943 Taxes (35%): 131.176.500 FY2000 Taxes: 176,788,443
After-Tax Profit: 329,178,057 After-Tax Profit: 243.613.500 2000 A-T Profit: 572,791,557
Fiscal Year 2001 (not actually included, but presumably)
P & G-FSC P&G Total
Sales Income: Sales Income: $0
Costs (COGS): 374,790,000 COGS: 359,344,974
Costs (other): 2,721,689 Costs (other): 0
Profit: (377,511,689) Profit: (359,344,974) FY2001 Profit: ($736,856,663)
Taxes (12.17%): ( 45,943,173) Taxes (35%): (125,770,741) FY2001 Taxes: (171,713,913)
After-Tax Profit: (331,568,516) After-Tax Profit: (233,574,233) 2001 A-T Profit: (565,142,750)
Combined 2000-2001 Total
Profit: $ 12,723,337
Taxes: 5,074,530
A-T Profit: 7,648,807
The first, most important, thing to notice about this result is that P & G’s tax liability is actually more than it would be if P&G had not involved the FSC at all; that is, P & G’s tax liability increased by $621,362 (from $4.45m to $5.07m, or 14%). Correspondingly, P & G’s after-tax profit decreased under the IRS’s calculation by 7.5% (from $8.27m to $ 7.65m). The second thing to notice is that the IRS used the wrong tax rate for the FSC. Pursuant to 26 U.S.C. § 291(a)(4)(A), when using the “arm’s length” middle-man approach, § 925(a)(3), a company can exempt 30% of its profit from taxation, which results in 70% of the standard rate (35%), which is 24.5%. The IRS used the 12.17% FSC tax rate that only applies to the pass-through or CTI partnership approaches, which the IRS had expressly deemed unavailable. *480Ironically, the correct tax rate (24.5%) produces slightly more favorable results for P & G: $4.74m in tax liability and $7.98m in after tax profit.
Perhaps recognizing that its view of the FSC Program would be worse for P & G than no FSC Program at all, the IRS offered an “alternative” position, pursuant to IRS Internal Memorandum 4.60.9.5, in which it offered its interpretation and application of the CTI partnership approach:
Step 1: Estimate costs to determine P & G’s portion of the profit:
Advance Payment Transaction — P & G+P & G-FSC combined entity
Sales Income: $374,790,000
Estimated Costs: 362,066,663 (the IRS offered no explanation for this amount)
Combined Taxable Income (CTI): 12,723,337
P & G-FSC’s portion of CTI (23%): 2,926,368
Step 2: Use the Sales and Profit in Step 1 to back-calculate P & G-FSC’s costs, which is also P & G’s sales:
Fiscal Year 2000
P & G-FSC P&G Total
Sales Income: $374,790,000 Income [ = FSC’s cost]: 371,863,632
Costs [SI — profit]: 371,863,632 Costs: 0
$374,790,000 Profit [from Step 1]: $ 2,926,368 Profit [I — Costs]: $371,863,632 Total
130,508,410 Taxes (12.17%): 356,139 Taxes (35%): 130,152,271 Total Taxes:
244,281,590 After-Tax Profit: 2,570,229 After-Tax Profit: 241,711,361 A-T Profit:
Step 3: Finish the calculation:
Fiscal Year 2001
P& G-FSC P&G Total
Sales Income: Sales Income:
Costs: 2,721,689 Costs: 359,344,974
Profit: ($2,721,689) Profit: ($359,344,974) Total ($362,066,663)
Taxes (12.17%): ( 331,230) Taxes (35%): ( 125,770,741) Total Taxes: ( 126,101,970)
After-Tax Profit: ( 2,390,459) After-Tax Profit: ( 233,574,233) A-T Profit: ( 235,964,693)
Combined 2000-2001 Total $ 12,723,337
Total Taxes: 4,406,440
A-T Profit: 8,316,897
So, under this interpretation and application, the inclusion of the FSC creates a slight benefit for P & G — tax liability is reduced by $46,728 (from $4.45m to $4.41m, or 1%) and the total after-tax profit is correspondingly increased by $46,728 (from $8.27m to $8.S2m, or 0.6%) — but this benefit is far below any of those we calculated under our interpretation of the FSC Program.
E. Summary of the Competing Results
Based on the foregoing, we have eight competing results; four that we calculated, two that P&G calculated, and two that the IRS calculated. Here is a summary of those results:
Without any FSC Pass-through CTI Partnership Middleman
Taxable Income: $12,723,337 $12,723,337 $12,723,337 $12,723,337
Taxes: 4.453.168 2,887,605 3,785,193 4,060,498
After-tax Profit: Change from no FSC: 8.270.169 0 9,835,732 1,565,563 8,938,144 667,975 8,662,839 392,670
Percent improvement: 0% 18.9% 8.1% 4.7%
P & G’s Approach P & G’s (corrected) IRS’s Primary IRS’s Alternative
*481Taxable Income: $15,445,026 $12,723,337 $12,723,337 $12,723,337
Taxes: (80,144,132) (14,602,052) 5,074,530 4,406,440
After-tax Profit: 95,589,158 27,325,389 7,648,807 8,316,897
Change from no FSC 87,318,989 19,055,220 (621,362) 46,728
Percent improvement: 1,0 230% (7.6%) 0.6%
Having set the stage, we can now turn our attention to the actual dispute. P & G contends that its approach is correct and, if not, that it is entitled to an alternative approach of its choosing. The IRS contends that P & G’s approach, though facially acceptable, actually results in a “material distortion of income” and therefore the IRS may select the approach and decide how to apply it. And the district court found that P & G had misapplied the CTI approach (by failing to match costs with income) and that the “variance doctrine” barred P & G from an alternative approach of its choosing.
III.
As stated at the outset, we are presented with four sequential questions: (1) did the district court err in its analysis; (2) did P & G misapply § 925, (8) did the IRS improperly impose its own FSC method (in violation of its own regulations), and (4) does the variance doctrine apply?
A.
The first question concerns the district court’s proclamation that the CTI calculation for a given tax year requires a matching of all incomes and expenses for any (every) included transaction; i.e., that the CTI provision does not permit accrual-based accounting. This is not apparent in the plain language of either the statute or the regulation, see 26 U.S.C. § 925(a)(2) and 26 C.F.R. § 1.925(a)-lT(c)(6), nor — as the calculations in Section II.B.2 make evident — is it even necessary; the use of accrual-based accounting does not change the results of a proper CTI calculation.
The district court relied on General Dynamics Corp. v. C.I.R., 108 T.C. 107 (U.S. Tax Court 1997), but that reliance was misplaced. In fact, the issues, arguments, and analysis in General Dynamics are so different from the present case that the two are virtually opposites. In General Dynamics the taxpayer company proposed an interpretation and application of the completed-contract method of accounting by which it could completely omit or exclude certain expenses (i.e., prior “period costs”) from its CTI calculation, thereby creating a “double or extra benefit.” Id. at 128. In the present case, P & G has simply invoked the accrual-based method of accounting (as opposed to the completed-contract method) and has not attempted to omit or exclude any costs, but simply has reported them in accordance with established accrual-based-accounting standards.
In rejecting General Dynamics’s proposed CTI calculation under the completed-contract method of accounting (i.e., its omission of the prior period costs), the tax court explained:
The completed contract method requires income and deductions from long-term contracts to be reported in the year in which the contracts are completed.... This method of accounting provides an alternative to the annual accrual method of accounting for long-term contracts for which the ultimate profit or loss is not ascertainable until the contract is completed. The [completed-contract] method allows a taxpayer to account for the entire result of a long-term contract at one time. The purpose of the completed contract method is to match the costs of generating income with the income produced. In this case, however, petitioners try to use the completed contract method to avoid the matching of costs with income from export sales for pur*482poses of computing CTI as required by the regulations under sections 994 and 925. As a result, petitioners did not subtract all the costs related to their export sales as defined in section 1.994-l(c)(6)(iii), Income Tax Regs., from the export income that the expenditures generated.
Id. at 126-27 (certain citations and paragraph break omitted). Nowhere in its opinion does the tax court suggest that the accrual-based method (or “annual accrual” method) of accounting is impermissible, or that the taxpayer must estimate as-of-yet unfulfilled or unrecorded expenses so as to force the CTI calculation into a single tax year in which it “matches” all income and expenses.
Rather, the court held that — when proceeding under the completed-contract method, as opposed to the annual-accrual method — the taxpayer must include in its CTI calculation all of the recorded payments and costs (incomes and expenses) associated with the FSC transaction (i.e., the completed contract), and that includes prior period costs associated with that transaction even if the (domestic) parent company already recorded those costs on its own tax returns in the (prior) years in which they occurred. Id. Thus, General Dynamics is inapposite to the present issue.19
The district court was incorrect and its reasoning unsupportable. We must therefore consider P & G’s claims anew, and next assess P & G’s interpretation and application of 26 U.S.C. § 925.
B.
The next question concerns P & G’s interpretation and application of the partnership-method transfer-price provision, 26 U.S.C. § 925(a)(2), and the underlying CTI regulation, 26 C.F.R. § 1.925(a)-lT(e)(6). Although P & G and P & G-FSC claimed to have calculated the transfer price in accordance with § 925(a)(2) and § 1.925(a)-lT(c)(6), they actually did no such thing. In reality, P & G-FSC simply paid P & G a “transfer price” of 77% of the payment from P & G-Canada; that is, P & G-FSC paid P & G $288.59m (which is 77% of $374.79m, P & G-FSC’s gross receipts from P & G-Canada) even though P & G’s costs were $359.34m. See Section II.C, supra (the corrected version of P & G’s calculations). This resulted in a $83.48m gain for P & G-FSC and a $70.76m loss for P & G on a $12.72m overall pre-tax profit. And, because P & G is taxed at almost three times P & G-FSC’s rate (35% to 12.17%), P & G’s tax benefit (i.e., the tax deduction on its $70.76m loss) accrues nearly three times faster than P & G-FSC’s tax liability (he., the tax due on its $83.43m gain). The net result was a $14.60m tax benefit on a $12.72m profit, leading to a $27.33m after-tax (benefit) profit.
But the statute and regulation describe a partnership in which the Parent-I-FSC *483joint entity (i.e., P & G+P & G-FSC) calculates the total combined profit (i.e., the payment received from P & G-Canada minus P & G’s costs and P & G-FSC’s costs) and apportions that profit between the partners (23% to the FSC, 77% to the parent). After apportionment, the two entities calculate their tax liability separately, on their respective profits at their respective tax rates. See Section II.B.B, supra.
To restate the error in P & G’s interpretation and application: P & G did not apportion anything. That is, instead of apportioning 23% of the profit to P & G-FSC (as would have been proper), P & G diverted 23% of the gross receipts to P & G-FSC as profit. See Section II.C, supra (corrected version). Consequently, P & G’s actual calculation of its transfer price can be depicted in one of two ways, neither of which is allowed: (1) P & G performed a gross-receipts, pass-through calculation, akin to § 925(a)(1), using 23% instead of 1.83%; or (2) P & G performed an arm’s-length, middleman calculation, akin to § 925(a)(3), using an “actual price” between P & G and P & G-FSC of $288.59m. Neither the statute nor the regulation supports either of these approaches.
The IRS deemed this result a “material distortion of income” in violation of the controlling regulation, see 26 C.F.R. § 1.925(a) — lT(c)(6)(iii)(B) (“the FSC may not choose a method of accounting which, when applied to transactions between the FSC and [the parent company], will result in a material distortion of the income of the FSC or of [the parent company]”), and disallowed it. Because the undisputed overall pre-tax profit was $12.72m, yet P & G’s method of accounting produced an $83.48m gain for P & G-FSC, a $70.76m loss for P & G, and a $14.60 tax benefit (leading to a $27.33m after-tax profit), it is not unreasonable to label this a “material distortion of income.”
Regardless of whether these results actually depict a “material distortion of income,” we conclude that P & G’s (and P & G-FSC’s) interpretation and application of § 925(a)(2) was incorrect and, consequently, its results are unsupportable. We must therefore conclude that P & G did misapply § 925; reject P & G’s claim for a refund on this basis; and proceed to the third question, namely, whether the IRS improperly imposed its own method of accounting.
C.
Upon pronouncing P & G’s interpretation and application of § 925(a) “impermissible,” as leading to a “material distortion of income,” the IRS asserted that the consequence of that misinterpretation and misapplication was that § 925(a)(1) and (2) (i.e., the pass-through and the partnership approaches) were no longer available to P & G. Instead, P & G was bound to the IRS’s calculation under an independent-middle-man approach, i.e., § 925(a)(3). P & G replied that the IRS had no authority to impose such consequence, particularly in light of the regulation, which says:
[S]ection 925 permits [the parent company] to determine the allowable transfer price charged the FSC ... by its choice of the three transfer pricing methods ...: The [pass-through] method and the [partnership] method ... of section 925(a)(1) and (2), respectively, and the [independent middle man] method of section 925(a)(3).... If either [the pass-through or partnership method] is applied to a transaction, the [IRS] may not make distributions, apportionments, or allocations as provided by section 482 [i.e., the ‘arm’s length’ standard] and the regulations under that section.
26 C.F.R. § 1.925(a)-lT(a)(l). This appears to us to be dispositive.
*484The IRS had no statutory or regulatory authority to impose its own “arm’s length” calculation on P & G and P & G-FSC in the context of this dispute and, in fact, appears to have been forbidden from doing so. So we answer the third question in the affirmative. The IRS contends, however, that, because this is a taxpayer claim for a tax refund, this error is essentially irrelevant because the taxpayers (P & G and P & G-FSC) are limited to the arguments they raised and the relief they sought in the underlying administrative proceedings. And this takes us to our fourth question.
D.
After the district court granted summary judgment to the IRS — rejecting P & G’s interpretation and application of § 925(a)(2) and upholding the IRS’s calculation of P & G’s tax liability based on its “arm’s length” calculation — P & G moved the court for a clarification or modification of the order. P & G sought to recalculate its tax liability using the pass-through method, as that would be more favorable than the IRS’s calculation. See Section II.E, supra.
The IRS opposed P & G’s motion based on the variance doctrine. See Estate of Bird, 534 F.2d 1214, 1219 (6th Cir.1976) (“[T]he grounds on which a claim for a tax refund is made must be specifically set forth in the claim for refund itself, otherwise the court in a refund action is without jurisdiction to consider them.”). The district court agreed, concluding that neither P & G nor P & GFSC had set forth a pass-through alternative (i.e., 1.83% of gross-receipts) in their refund claims.
The problem with this conclusion is that both P & G and P & G-FSC did set forth this argument in their refund claims, stating that, given the IRS’s alternative calculation (based on its new CTI result), “the correct pricing method that would then apply would be the pricing determined under the 1.83% gross receipts method of IRC § 925, which would result in a decrease in income to [P & G] and an increase of income to P & G-FSC in an amount the [IRS] has not calculated.”
The district court erred by denying P & G’s motion based on the variance doctrine, under these facts. P & G is entitled to recalculate its tax liability under the pass-through method.
IV.
For the foregoing reasons, we REVERSE the district court’s grant of summary judgment in favor of the IRS and REMAND for further proceedings consistent with this opinion.
KETHLEDGE, Circuit Judge,. The statutory provisions that govern this case, 26 U.S.C. §§ 921-927, were repealed November 15, 2000. See "FSC Repeal and Extraterritorial Income Exclusion Act of 2000,” Pub.L. 106-519, § 2, 114 Stat. 2423. But, under the phasing out provision, a FSC such as the present P & G-FSC, which was in existence as of September 30, 2000, could continue until January 21, 2002. See id. at § 5(c).
. In this transaction, the "buyers” (P & G-Canada and P & G-FSC) paid for the goods in advance (in 2000) for delivery later (in 2001). Under traditional accrual-based accounting methods, a company reports income (payment) in the tax year in which it is received, see 26 U.S.C. § 451(a), and reports expense (cost) in the tax year in which it is actually incurred, see 26 U.S.C. § 461(h)(2)(A)(ii). Thus, for purposes of calculating the tax liability associated with this "advance payment” transaction, P & G-FSC and P & G recorded income in 2000 and expenses in 2001.
. Cost of Goods Sold ("COGS”).
. P&G filed its own tax returns and separate returns for P & G-FSC, its wholly-owned subsidiary. Those returns are at issue in this case. The IRS did not challenge P & G-Canada's tax returns, so those are not at issue here.
. Under the FSC Program, the FSC could exempt 15/23 of its profit from taxation under either of two specially created approaches, namely § 925(a)(1) and (a)(2). See 26 U.S.C. § 291(a)(4)(B) (effective Aug. 20, 1996 to Dec. 28, 2007). Exempting 15/23 means paying tax on 8/23, which is the same as paying 8/23 as much tax on the entire profit. This is represented by altering the FSC's effective tax rate to 8/23 of the standard rate (35%), which is 12.17%.
For the third possible approach (§ 925(a)(3)), the FSC Program allowed the FSC to exempt 30% of its profit from taxation. See 26 U.S.C. § 291(a)(4)(A). This results in 70% of the standard rate (35%), which is 24.5%.
. Contrary to the misapprehension in the concurring opinion, there is no “assumption" here. This is a pure calculation and our omission of a stated “transfer price” does not render it less so or necessitate any assumption. In mathematics, the same equation can be represented in a variety of ways and still yield the same (correct) result. In this case, we chose the most direct route of calculating the end result (i.e., the profit subject to tax); one that did not require specific calculation of the underlying "transfer price.”
Lest it appear that something is amiss, let us be very clear that the underlying “transfer price” is easily calculated by application of basic profit-loss calculations: Profit = Income — Cost. Or, to state it more specifically for this particular situation:
FSC Profit = FSC Income — FSC Incidental Costs — Transfer Price.
FSC Profit must equal 23% of CTI, or $2,926,368 (as calculated in the first six lines of the above table). FSC Income is given: $374,790,000. As is FSC Incidental Cost: $2,721,689. Therefore, the profit-loss equation looks like this:
FSC Profit = FSC Income — FSC Incidental Costs — Transfer Price.
$2,926,368 = $374,790,000-$2,721,689-Transfer Price
Solving the equation gives us a Transfer Price of $369,141,943. We can, of course, check this result by inserting it into the profit-loss calculation for the parent, P&G:
P & G Profit = Transfer Price — P & G Costs.
P & G Profit = $369,141,943-$359,344,974 = $9,796,969.
This is the same profit we calculated for P & G in the seventh line of the table above.
The next footnote contains the same calculation for the accrual-based method and, as will be seen, it is not — as the concurrence contends — "more complicated” though it is a bit more cumbersome, in that we have to do it twice.
. It is easy to calculate a "transfer price” in this accrual-based-accounting scenario by following the same basic profit-loss equation that we used in the prior footnote: Profit = Income — Cost.
For tax year 2000, we calculated the CTI as $372,068,311, so 23% of that is $85,575,712 = 2000 FSC Profit.
Tax Year 2000 FSC Profit = FSC Income— FSC Incidental Costs — Transfer Price. $85,575,712 = $374,790,000-$2,721,689-Transfer Price.
Solving the equation gives us a Transfer Price of $286,492,599 for Tax Year 2000. We can check this result by inserting it into the profit-loss calculation for the parent, P&G:
2000 P&G Profit = Transfer Price — P & G Costs.
2000 P&G Profit = $286,492,599-$0 = $286,492,599.
This is the same profit we calculated for P & G in the table above.
It is just as easy to calculate the "transfer price” for tax year 2001. In tax year 2001, the CTI is ($359,344,974), and 23% of that amount is ($82,649,344), which is the 2001 FSC Profit:
Tax Year 2001 FSC Profit = FSC Income— FSC Incidental Costs — Transfer Price.
($82,649,344) = $0-$0 — Transfer Price. Solving the equation gives us a Transfer Price of $82,649,344. We can check this result by inserting it into the profit-loss calculation for the parent, P&G:
2001 P&G Profit = Transfer Price — P & G Costs.
2001 P&G Profit = $82,649,344-$359,344,974 = ($276,695,630).
This is the same profit we calculated in the table above. Nothing about this is "complicated” or "assumed.”
. The cited statutory provision is commonly referred to as the “arm’s length standard” for "allocation of income and deductions among taxpayers,” and states in pertinent part:
In any case of two or more organizations ... owned or controlled ... by the same interests, the [IRS] may ... allocate gross income, deductions, credits, or allowances between or among such organizations, ... if [the IRS] determines that such ... allocation is necessary in order to prevent evasion of taxes or clearly to reflect the income of any of such organizations....
26 U.S.C. § 482.
. The IRS also proposed an "alternative” position, in which it tendered its (incorrect) application of § 925(a)(2)'s "combined taxable income” calculation. These positions are discussed in Section II.D, infra.
. Under this "arm’s length” middle-man approach, § 925(a)(3), a company could exempt 30% of its profit from taxation. See 26 U.S.C. § 291(a)(4)(A). This results in 70% of the standard rate (35%), which is 24.5%.
.The tax code's no-loss-rule provision states in full:
Limitation on FSC income {‘no loss' rules). If there is a combined loss on a transaction or group of transactions, a FSC may not earn a profit under either the combined taxable income method or the gross re*477ceipts method. Also, for FSC taxable years beginning after December 31, 1986, in applying the gross receipts method, the FSC’s profit may not exceed 100% of full costing combined taxable income determined under the full costing method of § 1.925(a)-lT(c)(3) and (6). This rule prevents pricing at a loss to the related supplier. The related supplier may in all situations set a transfer price or rental payment or pay a commission in an amount that will allow the FSC to recover an amount not in excess of its costs, if any, even if to do so would create, or increase, a loss in the related supplier.
26 C.F.R. § 1.925(a) — IT(e)(l)(i).
. This "income” to P & G-FSC in 2001 is the "loss reimbursement” from P&G, made pursuant to P & G's interpretation of the "no loss rule.” Similarly, P & G’s "Costs (other)” in 2001 represents this "loss reimbursement.”
. Recall that P & G-FSC reported $2,721,689 in "costs (other)” on its returns for tax year 2000. For further information, please refer to the first table in this opinion.
. Imagine that you shift money from one hand to the other and back again, and that *478when you shift it to the one hand you tax it and when you shift it back you "un-tax” it. If both hands had equal tax rates, this circular shifting would have no effect. But if they had different rates, you would end up with more (or less) money at the end of the cycle.
That is, if you shifted money to one hand and taxed it at a low rate, and then shifted it back and "un-taxed” it at a high rate, then your tax would be low, your "un-tax” would be high, and you would end up with more money than you started with, even though all you did was shift it from one hand to the other (and back). That's the scenario.
. We have chosen to omit this calculation from our comparison. Our objective is to obtain consistent calculations, so that our comparison is meaningful. Because this calculation is circular and ultimately inconsequential (i.e., "meaningless”), we could just as accurately have added this calculation to all of our other approaches and created a consistent comparison in that way. We have chosen to omit the calculation from this approach because omitting it provides for an easier (and cleaner) calculation, thus making omission more practical than widespread addition.
. Section 446(c), "General Rule for Methods of Accounting,” says:
Permissible methods. — Subject to the provisions of subsections (a) and (b), a taxpayer *479may compute taxable income under any of the following methods of accounting—
(1) the cash receipts and disbursements method;
(2) an accrual method;
(3) any other method permitted by this chapter; or
(4) any combination of the foregoing methods permitted under regulations prescribed by the Secretary.
26 U.S.C. § 446(c).
. "Except as otherwise expressly provided in this chapter, a taxpayer who changes the method of accounting on the basis of which he regularly computes his income in keeping his books shall, before computing his taxable income under the new method, secure the consent of the Secretary.” 26 U.S.C. 446(e).
. The concurrence would hold that General Dynamics is wrong rather than distinguishable (for, if it is correct and indistinguishable, then why would we decline to follow it?). The concurrence surmises that “total costs” must mean the same thing under both accounting methods and concludes that "it seems to me that 'total costs' would fairly include costs incurred and deducted in a later tax year.” This begs the question of how the parent (or FSC) could include costs that have not yet arisen (i.e., “costs incurred and deducted in a later tax year”). The concurrence offers no answer to this question, which is a difficult question if one assumes — as the concurrence does — that the differing accounting methods are indistinguishable. But we need not answer that question because we conclude that the two accounting methods are distinguishable, and General Dynamics applies to one but not the other, so General Dynamics is distinguishable.