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Recently there has been quite the uproar in even the national news media about certain companies (usually high tech ones) “backdating” grants of options made to their executives. If such an allegation was true this would mean that the people granting the options selected the dates the options were granted to provide the best (most lucrative) reward to those receiving the grants and this would have ti have been done “after the fact”.
The issue was covered in recent Wall Street Journal (WSJ) articles (see this link http://www.biz.uiowa.edu/faculty/elie/wsj2.htm for one such article explaining how they reached their conclusions). When recent articles hit the (San Jose CA) Bay area “silicon valley” papers I took notice especially as one of the firms named (KLA Tencor ticker KLAC) was one I had worked for at one time.
The Sunday (6/4/2006) issue of the San Jose Mercury News (SJMN) contained a front page article addressing the “options backdating” issue. This article was of particular interest to me as there was actually some data to work with. So of course I attempted my own cut at a “Wall Street Journal likelihood of all of those Ken Levy (KLA Tencor's chairman) grants being given on these exact days of very low strike prices” probability analysis. This was a very interesting and simple analysis to do.
To summarize … my results show that for the five (out of ten) Ken Levy grants about which information was provided in the SJMN article the likelihood of all of these grants falling on the low price days they did is on the order of 3.2e7 (one chance in 3.13 million). While my quickie result does not exactly match the WSJ result both analyses indicate a very unlikely occurrence. My analysis thereby confirms the conclusions reached by the researchers in the WSJ article while using a slightly different probability analysis method.
For completeness Sunday's SJMN article can be found on this link …
http://nl.newsbank.com/nlsearch/we/Archives?p_action=doc&p_docid=1121507B3376AEC0&p_docnum=1
However the web version of this article does not contain the price history data figures I used as raw material for my probability analysis. If this link does not work, go to the San Jose Mercury news website and then search their news archives for “KLA Tencor option backdating” – the story to look at was published on 6/4/06.
I will fully explain my analysis below. And list one caveat – the SJMN article has very low resolution plots and to arrive at my numerical information I had to do some scaling using these plots which certainly detracts from the exactness of my answer. However my result clearly shows that the timing of the grants was rather unlikely.
Here is how I arrived at that conclusion …
• Start first by looking at the SJMN article specifically the three yearly KLAC closing price history charts. On these graphs the days and prices of five of Ken Levy's ten grants are provided. Note that each plot starts at July 1st and ends on June 30th of the following year. These plots will be the working material for our analysis.
• Now determine the option grant prices and KLAC stock price range throughout each year … for Year 1 (9900) KLAC price ranged $31.18 to $97.65 and there was one grant (at $34.12); for Year 2 (0001) KLAC price ranged $27.06 to $65.88 and there were three grants (at $42.35, $27.06 and $31.76, $27.06 being the yearly minimum); and finally in Year 3 (0102) KLAC price ranged $29.41 to $70.00 and there was one grant (at $29.41, yearly minimum). The dollar figures above were arrived at by measuring and scaling from those low resolution graphs.
• Now we decide how to do our analysis – we will use a slightly different method than the WSJ did (they had a daily database so they looked for the performance of the stock in the 20 trading days immediately following the grant and compared that performance to the “20 days afterward” performance on all other trading days in the year and ranked the performances of each trading day from best to worst. So to simplify my life (I have no database, only those three low resolution graphs) I will just look for minimum closing prices, this information being far easier to scale off of the poor quality graphical data in the SJMN article). We will develop the probability of each award being at or below its price level (ranking each grant days performance) during the trading year it was issued and then find our final 5grant probability value by simply multiplying all award probability values together (i.e., each award was assumed to be probabilistically independent of the others).
• Start with Year 1 (99 – 00) … (single) option award price was $34.12 and yearly minimum price was $31.18. There were approximately 75 trading days in which KLACs price ranged between these two values and on 64% of those days (48 trading days) KLACs price was below the award price but above the minimum price (this percentage is found by scaling and I caution you does not provide an exact result with the low resolution of the graph). With a total of 252 trading days a year our “award probability” (probability of the award being at or below the strike price level) is just 48/252 (the award was given on the 48th lowest price day of the year). This is a simple analysis as we are merely counting days here. And then drawing our conclusions.
• Now in Year 2 (00  01) … award prices were at $42.35, $27.06 and $31.76 and yearly minimum price was $27.06. By following the procedure outlined immediately above (for the first and third grant only as the second grant was at the yearly minimum price so we know its award probability by inspection) we find the individual grant “award probabilities” to be 165/252 (for the first grant at $42.35), 1/252 (for the 2nd grant at $27.06 which after all was at the lowest price of the year) and 41/252 (for the 3rd grant at $31.76)
• Similarly, in Year 3 (01  02) the sole option award price was $29.41 and this was also the yearly minimum stock price. So very straightforwardly the option's “award probability” was 1/252 since it was at the lowest price of the 252 trading day year.
• Now calculate our final answer (the probability of getting all five of these option awards) by simply multiplying together each of the five separate individual grant “award probabilities” determined above …
P(Levy's 5 grants) = (48/252) * (165/252 * 1/252 * 41/252) * (1/252) = 3.19e7
This is a very low probability indeed. Note that this is just one chance in 3.13 million.
The probability analysis is just that simple. We have confirmed herein the WSJ's conclusion that these grants were very unlikely to be given purely randomly, (because if the awards were purely random you ought to get options on both high and low price days, right?).
Recall that the WSJ calculated a 1 in 100 million probability for the entirety of Ken Levy's 10 grants and as I had data on only 5 of these grants I worked only with that data.
Where will this go from here? The SEC will be continuing to investigate and hunting through the paperwork. I would imagine that there will be an eventual monetary settlement but what I do not know about how the SEC works would fill volumes. I would guess that explaining a probability analysis in court to fully convince a jury might be very difficult.
Finally  sometimes the very dry subject of probability can be heck of a lot of fun!
That's all folks.
senegalP
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