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Foreword

First things first: I have owned shares of Yongye International (NASDAQ:YONG) since the fall of 2009.

I apologize for the fact that this rebuttal is way overdue. Back in early May, after reading Mr. Roe’s May 5th article Yongye's Top Customers: Discrepancies and Inconsistencies“ published by Seeking Alpha, I thought that somebody (meaning absolutely, definitely somebody other than me) will point out the illogic in certain claims by Mr. Roe. Did not happen. I also thought that the article would stay what it was, a minor annoyance. Did not happen. The article is now referenced in at least one class action filing. That made me think: In the time when we are calling more transparency from the companies we invest in, we seem to allow some distortion artists among us to convince the same companies that this transparency is not such a good thing. If honest reporting beyond what SEC requires gets twisted in ways companies will find expensive and time consuming to defend against. And that thought made me write this rebuttal. Let’s begin.

On May 5, 2011 Seeking Alpha published an article titled “Yongye's Top Customers: Discrepancies and Inconsistencies“ by one Richard X. Roe. In this article Mr. Roe makes several claims that can be proven to be unsupported by any fact available to Mr. Roe. Mr. Roe’s ingenious use of false logic and misleading math to build his case requires some coverage of related mathematics in order to expose what is so profoundly wrong with certain claims made by Mr. Roe.

Mr. Roe’s Claim # 2 Reproduced

Mr. Roe uses a particular approach to build several of his claims (claims 2, 10, 11, 17, 19 and 20). Out of these claims the claim # 2 is reproduced below, hopefully in a more readable form. All numbers in the claim refer to tables in Figure 1 and the tables themselves correspond to Yongye International SEC filings [1], [2] and [3]. In the tables category Other Provinces represent the whole of provinces having sales less than the sales in the smallest of top three provinces, category Other Customers is similarly the whole of customers having sales less than the smallest of top 5 customers. Note: in agreement with Mr. Roe’s claim #1 the sales in Xinjiang province are corrected to 47% of total sales and are now in line with the dollar figures given by Yongye.

Mr. Roe’s Claim: The top five customers represented $10,767,153 of sales in 2007 and no top five customer was in Inner Mongolia. Yongye’s total sales were $13,137,406. From the preceding it follows that Yongye’s sales in Inner-Mongolia must be less $2,370,253 ($13,137,406 - $10,767,153). This result contradicts Yongye’s reported sales of $2.7M for Inner-Mongolia, a discrepancy of at least $0.3M.

Some additional Information not provided by Mr. Roe but present in Yongye’s SEC filings [4] - Yongye sales presence: 10 provinces. Yongye distributor count: 18.

click to enlarge


Figure
1: Yongye Sales by Province and by Customer, 2007

From Sales Matrix to Sales Tables

Yongye International reports details about its sales using two types of tables: Sales by Province and Sales by Customer. In these tables Yongye International uses the word customer to indicate a province or county level distributor of Yongye’s fertilizer product. Province, obviously, is a Chinese administrative equivalent of US state. Sales information underlying these tables can be presented as a two dimensional matrix. Known provinces (here Xinjiang, Inner-Mongolia and Hebei) and the category Other Provinces are listed as horizontal dimension of the matrix. Known customers (Xinjiang, Beijing, Hebei, Dalian and Jiangsu) and the category Other Customers are listed as vertical dimension of the matrix. Resulting matrix has 24 (6x4) elements that cover sales by all distributors in all provinces – the sum of the elements is equal to total sales.

Yongye International does not provide sales information in this format, and with this detail one is reduced to guess possible values for each matrix element. What is known is the result of row-wise summation of the matrix elements: the Sales by Customer table. Column wise summation of the matrix elements produces the Sales by Province table. Two possible (guessed) distributions of Yongye’s sales are presented in Figures 2 and 3. Both distributions do produce the numbers reported by Yongye International. One can take this as an indication that Yongye’s reports are, at least, consistent.

However, both distributions do violate Mr. Roe’s assertion that the sales by top 5 customers must be all outside Inner-Mongolia. It is actually impossible to produce a sales distribution that would match the tables in Figure 1 and simultaneously satisfy Mr. Roe’s assertion (more about this later). This surprising fact does not prevent us from testing the generality of Mr. Roe’s method by creating an imaginary sales distribution that satisfies Mr. Roes stated restriction. The sales distribution created for that purpose is in Figure 4.


Figure 2: A Yongye Sales Distribution Matching the Tables in Figure 1


Figure 3: Another Yongye Sales Distribution Matching the Tables in Figure 1

Figure 4: Imaginary Sales Distribution Satisfying Mr. Roe’s Assertion

Repeating Mr. Roe’s calculation with the sales distribution in Figure 4 gives: <Total Sales> - <Top 5 Customers> = 13.14 M$ - 10.12 M$ = 3.02 M$ > 2.7 M$ (actual sales number for Inner-Mongolia in Figure 4). The discrepancy is now 0.32 M$ and in the opposite direction from the one calculated by Mr. Roe using Yongye’s reported sales numbers for 2007. It seems that Mr. Roe’s method has hard time matching the actual numbers.

Assumptions, Assertions and Many-to-One Mappings

Examination of the tables in Figure 1 reveals some facts about Yongye’s sales. The largest customers listed in the Sales by Customer table are identified by their home (official) location, which can be either a city or a province. Proof: Dalian is a large port city in Liaoning. The presence of Beijing (city or province, no matter) only in the Sales by Customer table makes it obvious that a customer can operate in multiple provinces, otherwise Beijing should be the second largest province in the Sales by Province table – it is not. Moreover, Yongye International does not say anywhere in the documents referenced here that the Beijing customer is an exception. It follows: one must assume that it is possible for any listed customer to have sales outside its home (official) province unless one can prove otherwise. There is no evidence contradicting this assumption.

The sales in Xinjiang province are greater than sales to any single customer, indicating that a province must have more than one distributor. This fact is also obvious when comparing the reported number of distributors, 18, to the number of provinces having Yongye’s sales presence, 10. Yongye International does not say anywhere in the documents reference here that the Xinjiang province is an exception. It follows: one must assume that is possible for any listed province to have more than one distributor operating within its borders unless one can prove otherwise. There is no evidence contradicting this assumption.

Mr. Roe’s assertion that the top five customers have all their sales outside Inner-Mongolia ignores the first assumption above. Maybe he sees in the tables something less obvious that allows him to do so. That possibility needs to be examined further.

In general any sales distribution by customer (distributor) and state (province) can be treated as a two dimensional table or matrix where each element ai,j in the matrix represent the sales by distributor i in province j, and therefore permissible values for element ai,j are zero (no sales) or some positive number (sales). As before, the sum of all elements in the matrix equals to total sales. Sales by Customer and Sales by Province tables are obtained using row and column wise summations of the elements in this matrix. A generalized sales distribution matrix using dimensions from Yongye’s sales tables is in Figure 5.

Figure 5: A Generalized Sales Distribution Matrix Having Dimensions Matching the Tables in Figure 1

In the row-wise summation of elements ai,j four members of the set {ai,j} are added to obtain one member of the set {bi} (Sales by Customer table). In set theory this is called many to one mapping. This mapping is non-reversible – if only set {bi} is given (the case with Yongye’s sales tables) then one cannot trace from any member of the set {bi} back to the elements of the set {ai,j} that formed the member of the set {bi}. The same mathematical fact holds for the column-wise summation of elements ai,j forming the members of the set { cj}, the Sales by Province table – non-reversible.

As shown above, the tables at hand cannot provide Mr. Roe the certainty needed for him to state “no top five customer was in Inner Mongolia” as a fact (assertion). Does Yongye International provide the backing information somewhere in the documents referenced here? Well, the company does not. Mr. Roe’s assertion is reduced to an assumption in dire need of proof.

Figure 6 below provides a simple illustration of some mappings defined in set theory. References [5] and [6] provide a short explanation of concepts used.

Figure 6: Examples of Mappings as Defined in Set Theory

Mr. Roe’s Assumptions and Some Set Theory

Previously it was stated that it is impossible to produce a sales distribution that would produce the tables in Figure 1 and also satisfy Mr. Roe’s assumption. To see why, one is better off treating Sales by Customer and Sales by Province tables as sets. Figure 7 presents both the set {bi} and the set {cj} in the form exposing their constructor elements from the set {ai,j}. (Persons having now an irresistible urge to learn more about set theory are advised to cautiously read references [7] and [8] although that is not really necessary for understanding what follows.)

Figure 7: Sales by Customer and Province Sets {bi} and {cj} showing the Underlying Elements of {ai,j}

Both sets in Figure 7 have the following properties:

1. Construction rule: Each set contains every non-zero element from the set {ai,j} once and only once. This comes from the fact that no actual sales can be omitted or duplicated. Otherwise there would be no true match to total sales. Here we don’t care about zero valued elements – they don’t change the value of summation.

  • It is impossible to add any new members, except elements having zero sales value, to either set without violating the construction rule. Why? Duplicated actual sales elements of the set {ai,j}.
  • It is impossible to remove any element, except elements having zero sales value, from either set without violating the construction rule. Why? Omitted actual elements of the set {ai,j}.
  • It is impossible to exchange any non-zero member bi of the set {bi} with any non-zero member cj of the set {cj} without violating the construction rule. Why? Duplicated and/or omitted actual sales elements of the set {ai,j}.

When Mr. Roe’s arithmetic operations are broken down to steps, one gets the following:

a. Subtracting the top five customers from total sales leaves only the category Other Customers – that is the member b6 in the set {bi}.

b. Next, Mr. Roe demands that this remaining member must be equal to sales in Inner-Mongolia: the member b6 of the set {bi} must be equal to the member c2 of the set {cj}. This requirement is presented below.

[a6,1 + a6,2 + a6,3 + a6,4] = [a1,2 + a2,2 + a3,2+ a4,2 + a5,2 + a6,2]

Mr. Roe’s assumes that the top five customers do not have sales in Inner-Mongolia. This leads to following: a 1,2 = a2,2 = a3,2 = a4,2 = a5,2 = 0. The above equality requirement is reduced to

[a6,1 + a6,2 + a6,3 + a6,4] = [a6,2]

For this equality to hold the elements a6,1, a6,3 and a6,4 must also be zero. This means that customers not in top 5 must confine their sales to Inner-Mongolia. Hopefully they obey in China.

Mr. Roe should have stated his ASSUMPTION as follows:

”If top five customers have all their sales outside Inner-Mongolia and the remaining customers have all their sales in Inner-Mongolia then and only then we can require that the sales in the category Other Customers equal to sales in the province of Inner-Mongolia.”

This means, by the way, that in Mr. Roe’s China there must be a horde of 13 distributors having a real sales melee in Inner-Mongolia while the top 5 distributors are tasked to take care of the remaining 9 provinces. Not likely. Now, back to why it is impossible to produce a sales distribution matching both the tables in Figure 1 and Mr. Roe’s assumptions. The Other Customers sales value in the tables is larger than the sales value in Inner-Mongolia and that immediately tells us that at least one of the top 5 customers must have sales in Inner-Mongolia. The question is: did Mr. Roe knowingly exploit this observation?

Considering all the effort Mr. Roe did put into making members of different sets equal, one can now wonder if Mr. Roe is comparing apples to oranges. To see if that is the case one needs yet another analysis tool: dimensional analysis.

Dimensional Analysis Says: “But Mr. Roe. It is Apples and Oranges!”

What is dimensional analysis? One (really minor) aspect of dimensional analysis is checking derived formulas and calculations using only units for the entities involved – a neat way to check if the formulas make sense or if the calculations have any change of being correct. References [9] and [10] contain discussion about dimensional analysis. In order to use dimensional analysis here, it is best to treat set elements ai,j , bi and cj as records containing units. After considering all the information provided by the tables in Figure 1 one will conclude that element ai,j can be seen containing following units of information.

Table Element

1st Dimension

2nd Dimension

3rd Dimension

4th Dimension

ai,j

value of sales

customer

province

year

Next, in order to make differences clear, we expand the province dimension to cover all provinces in China. There is nothing wrong or illegal with this expansion, it just introduces a number of zero valued elements in the distribution and does not alter any summation results (dimensional analysis here, not set theory). Elements bi are obtained by row-wise summation of elements ai,j (over all provinces in China). This gives for the units

Table Element

1st Dimension

2nd Dimension

3rd Dimension

4th Dimension

bi

value of sales

customer

China

year

Similarly, cj is a summation of elements ai,j over all customers. This gives for the units

Table Element

1st Dimension

2nd Dimension

3rd Dimension

4th Dimension

cj

value of sales

all customers

province

year

Dimensional analysis states that arithmetic operations between values in Sales by Customer and Sales by Province tables make sense only if the units for bi and cj are the same.

Are they?

Table Element

1st Dimension

2nd Dimension

3rd Dimension

4th Dimension

bi

value of sales

customer

China

year

cj

value of sales

all customers

province

year

Nope! They are not. Note that this result does not exclude comparisons that are basically logical. For instance: The fact that sales in Xinjiang province are greater than sales by any customer in Sales by Customer table implies that there must be more than one customer operating in Xinjiang – a logical comparison. But putting an equal sign between any member of Sales by Customer table and any member of Sales by Province table is an arithmetic operation. This means that Mr. Roe’s calculation for the claim #2 is void of any evidentiary value.

Mr. Roe Other Claims Based on Nothing

The above conclusion holds also for Mr. Roe’s claims 10, 11, 17, 19, 20. They all involve similar manipulations between Sales by Customer and Sales by Province tables. These claims are listed below; logic, math and dimensional violations are indicated by bold red font.

Claims 10 and 11:

(10) In 2008, customers who were not among the top five accounted for a total of $3,982,458 of sales ($48,092,271 - $44,109,813 ), per Item 7 and Item 9. There was only one top five customer in Xinjiang and it had $6,886,624 of sales, per Item 7. So, customer(s) in Xinjiang accounted for no more than $10,869,082 ($6,886,624 + $3,982,458) of sales. That number contradicts the $13,177,694 number for Xinjiang province in Item 10, a discrepancy of at least $2,308,612. Note also that the $13,177,694 number for Xinjiang province in Item 10 actually equals the sum of the sales of the top customers in Gansu and Xinjiang (see Item 7).

(11) According to Item 10, sales in Gansu province were $5,663,011 in 2008, however Item 7 shows that a customer in Gansu had $6,291,070 in sales, implying that some other customer(s) in Gansu had negative sales of $628,059.

Comment on claim 10: Guess where Mr. Roe moved his horde of distributors which was having such a party in Inner-Mongolia during 2007? Well, how about Xinjiang?

Comment on claim 11: This is a really brazen one – a direct arithmetic comparison between customer and province tables. Just a remainder: Mr. Roe does not have any means of knowing that the customer located in Gansu had its operations contained there.

Claim 17:

Item 7 shows that a customer in Xinjiang accounted for $6,886,624 of sales in 2008. Item 14 shows that the customer that accounted for $6,886,624 of sales in 2008 also accounted for $9,950,840 of sales in 2009. Therefore, a customer in Xinjiang accounted for $9,950,840 of sales in 2009, according to Item 7 and Item 14. Customer C from Item 14 is either in Xinjiang province or not. If Customer C from Item 14 were in Xinjiang, then sales in Xinjiang province would have been at least $26,695,996 in 2009 ($9,950,840 + $16,745,156), which contradicts Item 13's $16.7 million number for Xinjiang province. Alternatively, if Customer C from Item 14 were not in Xinjiang but in another province, then that other province would have accounted for at least $16,745,156 of sales, and thus, would have been a top three province, right after Hebei and Inner Mongolia, contradicting Item 13's placing Xinjiang, rather than that other province, as the third largest province by revenue in 2009. Therefore, Items 7, 13, and 14 cannot be reconciled.

Comment on claim 17: Complicated language but it still boils down to the fact that Mr. Roe is taking Sales by Customer table numbers and demanding them to be equal to selected Sales by Province table numbers. A BIG TIME NO-NO, as we already know.

Claim 19:

Item 18 shows that a customer had $5,209,290 of sales in the March 2010 quarter. However, according to Item 17 the top province in the quarter (Hebei) accounted for only $4.4 million, a discrepancy of $0.8 million.

Comment on claim 19: Well, what can we say except: “Here he goes again!”

Claim 20:

Item 18 shows that the top three customers accounted for 55% of sales (21% + 18% + 16%) in the March 2010 quarter, which exceeds the 41% number from Item 17 by 14.

Comment on claim 20: You guessed it; Item 17 is Sales by Province table. Did you really expect anything else?

Afterword

As for the rest of Mr. Roe’s article, I do recommend that my readers will re-read it with these comments in mind:

  • Claims 1, 9 and 18 are just complaining about typos.
  • Consider how a manufacture in the old, good USA can track its sales. How about by shipping addresses and distributor surveys? Which one do you think is more accurate? Then, how many times a year you think a manufacture will ask for distributor feedback? Every quarter? Twice a year? Once a year?

I hope I have managed to enlighten you about the quality of the subject article and also about the quality of its author.

Till next time.

References

1. Page 19, Yongye Prospectus filed 2008/09/12

2. Page 34, Yongye 10K filed 2009/03/24

3. Page F-18, Yongye 10K filed 2009/03/24

4. Pages 19 and 42, Yongye Prospectus filed 2008/09/12

5. Web page about Injection and Bijection: http://en.wikipedia.org/wiki/Bijection

6. Web page about Injection and Bijection: http://www.mathsisfun.com/sets/injective-surjective-bijective.html

7. Basic Concepts of Set Theory (.pdf), Functions and Relations, Lecture Notes, Barbara Partee, University of Massachusetts ,2006

8. Basic Set Theory (.pdf), Lecture Notes, James T. Smith, San Francisco State University, 2008

9. Lecture on Dimensional Analysis (.pdf), Murray S. Daw, Clemson University, 1999

10. The Physical Basis of Dimensional Analysis (.pdf), Ain A. Sonin, Massachusetts Institute of Technology, 2001

Source: Yongye International: A Victim of Bad Logic and Misleading Math