Virtually everyone interested in Herbalife (NYSE:HLF) is now familiar with Ackman's thesis that Herbalife is a pyramid scheme. One of Herbalife's key defenses is that every prospective distributor is provided with a "STATEMENT OF AVERAGE GROSS COMPENSATION OF U.S. SUPERVISORS", revised statement found on their website. Herbalife claims that this statement allows each prospective distributor to fully evaluate their prospects of making money, before signing up to become a Herbalife distributor.

There are three problems with this defense by Herbalife. One was addressed after a question by David Einhorn, namely that the gross compensation did not include expenses. Herbalife published a revised statement on 7/25/2012 noting that the gross compensation did not include expenses. One was highlighted by Ackman in his presentation that these numbers do not include about 93% of Herbalife distributors who make little or no money. This article is going to focus on the third problem - namely the numbers do not add up using math as we know it.

Please take a look at the revised statement below.

I want you to focus on two pairs of key numbers. Herbalife claims that about 39.4% of all leaders are "Active Leaders" defined as those who bought at least $2500 in product. Herbalife considers these as "sales" but as Ackman proved, Herbalife has no actual retail records of sales, so I'll refer to these as Sales Leader purchases from Herbalife. For these Active Leaders, Herbalife claims that they pay an average gross compensation of $7,354 with a median of $637.

For ALL sales leaders, both the Active Sales Leaders and the Inactive Sales Leaders, Herbalife claims that it pays an average compensation of $2900 with a median compensation of $741. If something doesn't seem quite right here about why ALL Sales Leaders would have HIGHER median gross compensation than the Active Leaders, read on...

**Let us do some math here. Real math, not Herbalife Magical Math.**

Let us take a representative sample of 10,000 Herbalife Sales Leaders. If you want to do the exercise with actual numbers of Total Sales Leaders, use as reported on page 5 of the 10-K filed on 2/21/2012.

*Table 1 - Figuring out Herbalife's compensation paid to InActive Distributors*

Total Gross Compensation paid to ALL 10,000 sales leaders @ average of $2900 | (10,000 X $2900) | $29,000,000 |

Total Gross Compensation paid to Active Leaders ONLY | (39.4% X 10,000 X $7,354) | $28,974,760 |

Total Compensation paid to Non Active Sales Leaders (Total Compensation Paid to ALL sales leaders - Total Compensation paid to Active Sales Leaders) | ($29,000,000 - $28,974,760) | $25,240 |

Total Number of Inactive Sales Leaders | (60.6% X 10,000) | 6,060 |

Actual Average Gross Compensation per Inactive Leader (Total Compensation paid to Non Active Sales Leaders / Total Number of Inactive Sales Leaders) | ($25,240/6060) | $4.16 |

**Now Let us Try to Square This With the Claim that The Median Gross Compensation for ALL Sales Leaders is $741.**

Now the Median Gross Compensation for ALL 10,000 Sales Leaders is obtained by ordering the Gross Compensation of each Sales Leader (Active and Inactive) by dollar amount and averaging the compensation of the 5000th and 5001st Sales Leader.

1,2,....5000, 5001,.....10,000

The average of the two according to Herbalife is $741.

Since the median is defined as the central ranked number, numbers above the median are greater than or equal to the median and numbers below the median are lesser than or equal to the median.Thus actual Gross compensation of every Sales Leader from 5001 to 10,000 is at least $741 (and probably more to account for the difference between the average of $2900 and the median of $741).

To try to make the numbers square let us make a couple of assumptions. Later on, I'll show how these assumptions have been made to try to support the numbers from Herbalife.

Assume the Sales Leaders from 0-5000 made $0 and the 5000th Sales Leader made $741 and the 5001st Sales Leader made 741.

Assume every Active Sales Leader makes more money than Any Inactive Sales Leader in the lowest 5000 Gross Compensation numbers.

Table 2. Calculating Minimum Amount of Money Herbalife Should Have Paid its Inactive Distributors

Minimum Total Amount paid to Inactive Sales Leaders from 5001-6060 | (1061 Inactive Sales Leaders X $741) | $785,460 |

**Checking The Effect of our Assumptions**

If you assume there are any number "n" Active Sales Leaders between 0-5000 then you would adjust the above calculation thus:

Table 3. Checking our Assumption that There are No Active Sales Leaders Ranked Below 5000

Minimum Total Amount paid to Inactive Sales Leaders from 5001-6060 | ([1061 Inactive Sales Leaders X 741] + ["n" more Inactive Leaders displaced from below rank 5000 X $741]) | $785,460 + 741n |

You can clearly see the minimum amount of money paid to the Inactive Sales Leaders is when n = 0 or that when you assume every Active Sales Leader makes more money than Any Inactive Sales Leader in the lowest 5000 ranked Gross Compensations.

If you assume that any Inactive Sales Leader between 0-5000 made any gross compensation "d" at all then the calculations would have to adjusted thus -

Table 4. Checking Our Assumption that Inactive Sales Leaders Ranked Below 5000 Made $0

Minimum Total Amount paid to Inactive Sales Leaders from 5001-6060 | ([1061 Inactive Sales Leaders X 741] + ["n" more Inactive Leaders ranked above 5000 X $741] + all the money "d" paid to Inactive Sales Leaders ranked below 5000) | $785,460 + 741n + Sum (all d) |

The total compensation paid to Inactive Sales Leaders is minimized when sum(all d) = 0 i.e. the Inactive Sales Leaders ranked between 0-5000 in gross compensation make no money at all.

**So We Arrive at this Conclusion...**

Now according to Herbalife's disclosures the total amount paid to ALL 6060 Inactive Sales Leaders, as calculated in Table 1, is **$25,240**

We have tried in every way to slant our assumptions to minimize the amount paid to the Inactive Sales Leaders. But we calculated the minimum amount that Herbalife should have paid ALL the 6060 Inactive Sales Leaders to achieve a median pay of $741 overall is at least $785,460. **By the rules of normal math $25,240 is not equal to $785,460.**

On page 5 of the 10-K filed by Herbalife on 2/21/2012, the total number of Sales Leaders is listed as 501,000 Active and Inactive as of December 31, 2011.

Herbalife has **overstated compensation to the Non Active Sales Leaders** **by at least $38.1 million (see Table 5).**

Table 5. Calculating the Total Overstatement of Payments to Inactive Sales Leaders

Shortfall in Payment to every 6,060 Sales Leaders (Active and InActive) [Calculated Total Minimum Payment fromTable 2 vs. Reported Total Minimum Payment Calculated in Table 1] | ($785,460 - $25,240) | $760,220 |

Shortfall in Payment Per InActive Sales Leaders (Shortfall in Table 2/ Number of InActive Leaders in 10,000 Sales Leader) | $760,220/6,060 | $125.45 |

Total Number of Sales Leaders (Active and InActive) | On page 5 of the 10-K filed by Herbalife on 2/21/2012 | 501,000 |

Number of InActive Sales Leaders | (60.6% x 501,000) | 303,606 |

Shortfall in payment to InActive Sales Leaders (Short fall in payment per InActive Trader X Total Number of InActive Traders) |
($125.45 X 303,606) | $38,087,022 |

Note that this is a lower bound, not a upper bound.

**Why is this number so important for a company with $3.5B in revenues?**

Let us read what Herbalife CEO Michael Johnson said about the "STATEMENT OF AVERAGE GROSS COMPENSATION OF U.S. SUPERVISORS" â€¦.

In other words, this is the document that Herbalife provides potential new victims before making them buy about $3000 worth of product to qualify as a Supervisor- excluding that the several thousand in other expenses that new supervisors mention, in the 91 pages of testimonials collected by Ackman (see here). Given that the average is skewed by the high earnings of the very few at the top, **the median is most relevant number to a potential recruit.**

**A Little More Math**

Let us calculate the best case scenario of what could be the median gross compensation for all 10,000 Non Sales Leaders

Table 6. Calculating the Maximum Amount the Median Value Could be For All Sales Leaders as Per Herbalife's Disclosures

The $25,240 was paid to the1061 Non Sales Leaders ranked from 5000-6060 |
(25,240/1061) | $23.78 |

Imagine you are a vulnerable, low information person trying to earn some money. You would look at Herbalife's "STATEMENT OF AVERAGE GROSS COMPENSATION OF U.S. SUPERVISORS". You would look at the median because you notice that the average is skewed by the few people at the top who make all the money.

You might think - well, if I'm successful enough to sell $2500 of product I could reasonably expect $637 in gross compensation and e*ven if I don't sell $2500 of product I could still expect about $741* as a median number in gross compensation. [Of course, being a low information, vulnerable mark, you would not realize in either eventuality that the expenses would leave you with a substantial loss].

The clue lies in the discrepancy that the median for ALL Leaders ($741) is very unlikely to be higher than the median for the higher earning subset of Active Leaders ($637) who buy at least $2500 in product from Herbalife.

In reality if you could actually work out the numbers from Herbalife's own filings, you would realize that you could not make a median of more than $23.78. **There's a big difference between a median Gross compensation of $741 and a median of just $23.78** when you are deciding whether to pay $3000 to become a "Supervisor". **Even you, a low information, vulnerable victim might just decide you won't risk $3000 to make $23.78 before expenses...**

The actual disclosure statement should read that the average gross compensation for ALL leaders is $2900 with the median being a number below $23.78. **The $741 median gross compensation number is flat out mathematically impossible.**

**The Big Takeaway**

This "STATEMENT OF AVERAGE GROSS COMPENSATION OF U.S. SUPERVISORS" disclosure has been repeatedly cited by Herbalife CEO Michael Johnson as a defense. It was publicly questioned by Einhorn on an earnings conference call and revised and refiled by Herbalife. How many potential distributors believed this document at face value? Herbalife owes investors and its current Sales Leaders a quick and prompt explanation.

**Disclosure: **I am short HLF. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.