This matter is before us on a taxpayer's petition to review a decision of the Tax Court redetermining income tax deficiencies and penalties for 1948, 1949, and 1950. The principal question we have to decide is this: Where the Tax Court has rejected the deficiency determination of the Commissioner of Internal Revenue, may it redetermine the deficiency by finding that 'it is not likely' that gross income exceeded a certain figure (substantially below that found by the Commissioner) and then utilize that figure as actual gross income because petitioner failed to extablish a lesser amount?1

During the years in question the petitioner, Lesly Cohen, operated the Kingston Club, a San Francisco card room. During the same period and using the same premises as his headquarters, Cohen operated as a so-called 'betting commissioner.' The latter activity was in violation of both state and local law. Either as a part of this latter activity (as appellant contends) or as a separate venture in which appellant engaged in personal betting (as the Tax Court found), Cohen also received income from a San Francisco establishment known as the Film Row Club.

As betting commissioner, Cohen would obtain opposite parties to wagers on horse races and other sporting events. Normally he did not accept a wager as 'placed' until he had found some other individual to 'lay off' the other side of the same event. When Cohen was able to 'lay off' the entire amount of the bet, his profit or loss would not depend upon the outcome of the event. Instead, it would be a fixed percentage or 'commission' of the total wager. When able to do so, Cohen would lay off the bet with his own local customers. When this could not be accomplished, he would lay off or cover the bet with other betting commissioners.

The commission to petitioner on bets handled for his own customers was five per cent on each bet, except that on horseracing bets only the loser paid a commission. On bets laid off with other betting commissioners, the commission was usually split. At times he found it necessary to waive his entire commission in order to get the bet laid off. Occasionally he was unable to lay off a bet and had to carry it himself, acting in such cases as a bookmaker.

This betting-commissioner enterprise was operated almost entirely on a credit basis. Ordinarily no money was posted with the petitioner in advance of the sporting event. Settlements with local bettors and other local commissioners were usually made in cash. Transactions with out-of-town commissioners were generally settled at periodic intervals by check. The latter settlements were in effect the balancing of accounts between Cohen and the out-of-town commissioners. They usually represented the net amount due from a number of bets rather than a single bet.

Petitioner maintained a 'revolving found' of about $3,000 is cash, which he used in making payments to local winners. The only cash deposits to petitioner's commercial bank account during the years in question were as follows: 1948-- $430; 1949-- $8,470; 1950-- $13,955. However, cash received from local bettors far exceeded these sums. Cohen's records of cash transactions as betting commissioner were kept only a few days until settlement was made. He never furnished to his accountant any records of his cash transactions or cash commissions received as betting commissioner. Accordingly, undeposited cash was not taken into consideration in preparing Cohen's income tax returns for those years.

The total deposits to Cohen's commercial bank account for the years in question were: 1948-- $508,384.23; 1949-- $404,118.69; 1950-- $283,129.80. These largely represented receipts from other betting commissioners in settlement of accounts. In addition to the checks so deposited Cohen received a large number of settlement checks which were endorsed by him but not deposited. The totals of these undeposited checks were: 1948-- $120,974.75; 1949-- $107,712; 1950-- $22,613.75.

Cohen made payments by check in settlement of accounts with out-of-town bettors totaling $292,283.46 in 1950. Payments in unspecified amounts were made under similar circumstances in 1948 and 1949.

During the three years in question Cohen did not maintain any permanent or detailed records or formal books of account pertaining to his betting-commissioner activities. He kept a daily 'master sheet' of transactions, but these were always destroyed a day or two later. It was therefore impossible to make an accurate determination of the amount of commissions received by Cohen.

An accountant employed by Cohen determined the latter's annual gross income as follows: The amount in the bank at the beginning of the year was subtracted from the amount at the end of the year. There was added to the resulting net increase or decrease in the bank balance all expenses of the cardroom business and all withdrawals made by or for Cohen. The result was considered petitioner's gross income. Underposited cash receipts and cash payments not substantiated by a memorandum wer disregarded. This was done on the theory that the $3,000 revolving fund remained approximately the same throughout the year.

Cohen's income tax returns were prepared by a certified public accountant on the basis of annual summary sheets supplied by the regularly employed accountant. The income tax disclosed and paid in Cohen's returns for 1948 and 1949 as amended was $13,863.65 and $19,190.51, respectively. His 1950 return disclosed that no tax was due, and none was paid for that year. These returns described Cohen's business as 'brokerage.'

Late in 1950, R. Parenti, an examining officer for the Internal Revenue Service, audited Cohen's income tax returns for 1948 and 1949. Basing his examination entirely on information furnished by Cohen's accountant, Parenti arrived at deficiencies of $5,505.67 for 1948, and $4,689.23 for 1949. In 1952, Glenn Adrian, another Internal Revenue Service agent, audited Cohen's tax return for 1950, and re-examined his returns for 1948 and 1949. At this time there was in progress a nationwide investigation of betting commissioners and others engaged in gambling activities. As a result, Adrian received from other revenue agents photostatic copies of checks which Cohen had received and endorsed but had not deposited to his account.

Adrian determined that all monies deposited in the commercial bank and all checks received and endorsed but not deposited and all wins from the Film Row Club constituted income. Because of lack of substantiation, no deductions were allowed for payouts or losses. All other deductions claimed by Cohen were allowed. Based upon Adrian's audit, Cohen was sent a notice of deficiency and penalty assessment for the years in question as follows:

                        Penalty
Year   Deficiency    50% for fraud      Total
-------------------------------------------------
1948  $  538,911.40    $269,455.70  $  808,367.10
1949     426,038.44     213,019.22     639,057.66
1950     228,561.34     114,280.67     342,842.01
      -------------  -------------  -------------
      $1,193,511.18    $596,755.59  $1,790,266.77

Cohen thereupon petitioned the Tax Court for a redetermination, asserting that there was no income tax deficiency and no penalty due for any of these years. In its unreported decision of December 12, 1957, now under review, the Tax Court determined the deficiencies and penalties as follows:

Year  Deficiency    Penalty       Total
------------------------------------------
1948  $ 72,164.36  $38,835.02  $110,999.38
1949    47,364.77   27,790.00    75,154.77
1950    49,004.79   24,502.40    73,507.19
      -----------  ----------  -----------
      $168,533.92  $91,124.42  $259,661.34

It will be observed that the Tax Court reduced the Commissioner's deficiency and penalty assessment for the three years from $1,790,266.77 to $259,661.34, or more than eighty-five per cent.

Two revisions in the Commissioner's determination apparently account for most of this reduction. One of these has to do with Cohen's asserted wins and losses in 1948 and 1949 from personal betting at the Film Row Club.2 His wins for these years at the Film Row Club aggregated $125,195.00, and his losses aggregated $148,987.50. The Commissioner included in income all gains from this source, but allowed no losses as balancing deductions. He disallowed losses on the ground that they were unsubstatiated, and also because the taxpayer was on a cash basis and had failed to show the year or years the losses were paid.

The Tax Court held that both of these grounds for disregarding losses were tenable and that the Commissioner's determination in this regard was therefore not arbitrary. Nevertheless it was held that the Commissioner had erred in failing to allow these gambling losses to the extent of gambling gains.3

The other major revision made by the Tax Court relates to the treatment of deposits in Cohen's commercial bank account. The Commissioner treated all such deposits as gross income and refused to allow any deductions for payouts. The Commissioner adopted this course because Cohen did not offer any substantiation of payouts, and did not maintain any records from which payouts could be calculated. Because of these circumstances the court held in effect that the Commissioner's treatment of bank deposits and his refusal to allow deductions for payouts were not 'arbitrary or invalid in whole or in part.'

The Tax Court nevertheless held that since these deposits were received in connection with Cohen's activities as a betting commissioner there must have been substantial payouts. Stating that the course adopted by the Commissioner in this regard was therefore 'harsh and unrealistic,' the Tax Court undertook to reconstruct Cohen's gross income on a more realistic basis.4

For the year 1948 the Tax Court first noted that Cohen's commercial deposits totaled over $500,000, of which only a little over $400 were cash deposits. The remaining deposits, it was concluded, largely represented the settlement of bets laid off with out-of-town commissioners, or the settlement of bets which they laid off with him. The credit balance could be in either direction and took into consideration wins, losses, and commissions. But these deposits, the Tax Court held, represented the settlement of accounts each involving more than one bet.

The court also took account of the fact that in 1948 Cohen received undeposited checks in excess of $120,000, which also represented settlements each envolving a number of bets. In addition, the Tax Court held, Cohen issued checks aggregating $290,000 to out-of-town commissioners representing settlements in their favor. Because each settlement usually represented several bets, the Tax Court decided that the total of lay-off bets must have materially exceeded the total of checks received, whether deposited, cashed, or endorsed, and checks issued.

Each of the out-of-town transactions referred to above, the Tax Court held, must have been counterbalanced by a local bet placed with or by Cohen. He had to split the commissions on the out-of-town portion of this business, but retained the full commissions on the local portion. In addition, the court determined, a substantial part of Cohen's business was the placing of local bets in which he represented both sides. On such business he received commissions from both sides, except on horse-race bets where only the loser pain a commission. The normal commission was five per cent, so when Cohen placed both sides of a nonhorse-race bet with his own customers he received ten per cent of the wager.5

Based upon the factors mentioned but without disclosing the arithmetic of its computation, the Tax Court determined that 'it is not likely' that Cohen received gross commissions as betting commissioner in 1948 in excess of $167,000. Holding that petitioner had failed to establish a lesser amount, the court accepted that figure as Cohen's 1948 gross income from this source. Subtracting the gross income of $56,795.13 of the Kingston Club, reported in Cohen's 1948 tax return, the court found a net understatement of income as betting commissioner in the amount of $110,204.87 for that year.

Similar calculations were made for 1949 and 1959, the court determining that the net understatements of income as betting commissioner were as follows: 1949-- $78,725.09; 1950-- $99,792.29. The figures for these three years are to be compared with the net understatements of income as determined by the Commissioner, which were: 1948-- $668,807.80; 1949-- $506,738.04; 1950-- $326,095.00.

It will be noted that in reconstructing gross income the method used by the Tax Court was to first determine the maximum amount of income it was likely that Cohen could have received. Then, in the absence of a showing by Cohen that he actually received a lesser amount, this maximum figure was accepted by the court as Cohen's gross income.

Petitioner argues that the Tax Court erred in thus charging him with the burden of proving that he owed no tax or owed a lesser amount than computed by the court. It is contended that such a course is permissible only where the Commissioner's determination is presumptively correct. Such a presumption does not exist here, petitioner asserts, in view of the uncontroverted facts and the Tax Court's rejection of the Commissioner's determination.

At the outset of a Tax Court proceeding to redetermine a tax deficiency, the Commissioner's determination is presumed to be correct. Thomas v. Commissioner, 6 Cir., 223 F.2d 83, 88. The burden of proof is thus placed upon the taxpayer to show that the Commissioner's determination is invalid.6

When the Commissioner's determination has been shown to be invalid, the Tax Court must redetermine the deficiency.7 The presumption as to the correctness of the Commissioner's determination is then out of the case. MacCrowe's Estate v. Commissioner, supra, 252 F.2d at page 294. The Commissioner and not the taxpayer then has the burden of proving whether any deficiency exists and if so the amount. It is not incumbent upon the taxpayer under these circumstances to prove that he owed no tax or the amount of the tax which he did owe.8

Applying these principles in the instant case, it must first be determined whether Cohen sustained his burden of proving that the Commissioner's determination was invalid.

The Tax Court found the Commissioner's method of determining deficiencies incorrect in two important respects. The Commissioner's method was also criticized as being 'harsh and unrealistic.' Making its own determination on a basis which made every possible assumption against Cohen, the Tax Court reduced the deficiency and penalty assessment for the three years from $1,790,000 to $260,000. The facts of record fully substantiate the Tax Court's criticsm of the Commissioner's method. They would have required that the Commissioner's determination be set aside even if the Tax Court had not done so.

The Commissioner's determination is invalid when it is arbitrary or erroneous. Greenwood v. Commissioner, 9 Cir., 134 F.2d 915, 919. Whether or not the Commissioner's determination in this case was arbitrary, it was at least erroneous in substantial respects. Cohen accordingly met his burden of proving that the Commissioner's determination was invalid. In view of these circumstances the further and inconsistent holding of the Tax Court to the effect that the Commissioner's determination was not invalid must be disregarded.

Respondent argues that the taxpayer did not meet his burden of showing that the Commissioner's determination, 'as redetermined by the Tax Court,' was arbitrary or invalid. The question, however, is not whether the determination was invalid after the Tax Court had reduced it by eighty-five per cent. The pertinent inquiry is whether it was invalid as originally computed by the Commissioner. See Helvering v. Taylor, supra, 293 U.S. at page 515, 55 S.Ct. at page 290. Respondent has not here claimed that the Commissioner's original determination was valid.

Cohen, having met his burden of proving the Commissioner's determination invalid, and thus taking the presumption of the correctness of that determination out of the case, was not chargeable with failure to make a further showing. Hence it was improper for the Tax Court in making its redetermination to accept its maximum possible gross income figure as the actual figure only because Cohen failed to show a lesser amount.

We neither state nor imply that the Tax Court's redetermination must be predicated upon precisely established income and expense figures. Since Cohan v. Commissioner, 2 Cir., 39 F.2d 540, it has been the recognized rule that absolute certainty is not required. It is sufficient if the Tax Court makes as close an approximation as it reasonably can. The Tax Court's permissible leeway in this regard is the same as that which the Commissioner originally had. See Bodoglau v. Commissioner, 7 Cir., 230 F.2d 336, 340.

In view of what has just been said, it may well be that the gross income figure accepted by the Tax Court will turn out to be supportable. We make no predictions as to this. But if it is supportable, it must be upon the basis of a finding of fact (which may be based upon approximations) as to gross income, and not upon the ground that the taxpayer failed to disprove a maximum figure selected by the Court.

For the reasons indicated it will be necessary to remand this case to the Tax Court for further proceedings. See Thomas v. Commissioner, supra. This being the case, it will serve no useful purpose to here consider in detail petitioner's contention that certain findings of fact are clearly erroneous and that in other respects they are inadequate for purposes of review. Two or three general observations, however, may prove helpful in the further proceedings.

When a petitoner in a tax case, or an appellant in any case, asserts that a particular finding of fact has no support in the evidence, the respondent or appellee, as the case may be, has the duty of calling to our attention any supporting evidence to be found in the record. This is true because of the practical difficulty which faces a petitioner or an appellant in trying to prove such a negative as compared to the ease with which the respondent or appellee can prove the affirmative by pointing to the supporting evidence. Here the respondent disagrees with petitioner's argument that several findings are clearly erroneous, but does not give us record references to the claimed supporting evidence.

Where the Tax Court makes a deficiency determination it must weigh all elements which are entitled to consideration. Bodoglau v. Commissioner, supra, 230 F.2d at page 340. But it is not sufficient to discuss these elements in a general way, as the Tax Court has done in the decision here under review. While it is permissible to employ approximations where this is necessary, the findings and decision should reveal the arithmetic employed by the court in reaching its final figure. That was not done here, and there is no way we can determine what weight was given to the factors discussed by the Tax Court in arriving at its determination.9

The Tax Court predicated its finding of fraud upon the dual prounds of substantial and consistent understatement of income and deliberate failure to maintain necessary records. These grounds are sufficient to support a finding of fraud if sustained in the evidence.10 Since at least the first of these grounds concerns a matter which will be under consideration in the remanded proceedings, we cannot now adjudicate this issue of the case.

Remanded for further proceedings.