In an interesting exercise, the market regulator's whole-time director Ananth Narayan computed the probability of a regular trader placing a buy/sell order just moments before such an order was placed by a big institutional client.
Narayan was adjudicating a case of front-running by a former chief dealer of IDBI Capital and Securities Markets, Gaurav Dedhia, and Dedhia's sister Kajal Savla.
The order issued by Securities and Exchange Board of India (Sebi) on May 31 records the regulator asking them to calculate this probability and Savla coming back with a submission which Narayan found "perfunctory, incorrect and misleading on several counts".
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Therefore, he calculated the probability of one of Savla's trades syncing with a big client's, in this case LIC's. His showed that the probability that on more than 15 occasions Savla buying a stock 3.5 minutes (as was seen) before the institution through the same brokerage does was 7.3 times in 1,000,000,000,000,000,000,000,000,000 (1 octillion or 1 followed by 27 zeroes).
As he commented later in the order, "the probability that the distinctive trading pattern and coincidences achieved by Noticee No.2 (Savla) during the investigation period is anything other than a pre-meditated scheme of front running, is a mind-numbingly low number that approximates to zero."
The Securities and Exchange Board of India (Sebi) asked Dedhia and Savla to disgorge over Rs 1.67 crore, which has been compted as wrongful gains made from front-running trades of institutional clients of the brokerage.
The brother and sister have been asked to disgorge this amount plus an interest of 12 percent per annum starting from June 14, 2022, which was the last date of front-run trades. They have also been fined Rs 25 lakh and Rs 15 lakh each.
The Securities and Exchange Board of India (Sebi) issued the order on May 31.
The big clients whose trades former dealer Gaurav Dhedia and his sister Kajal Savla front-ran include Life Insurance Corporation of India (LIC), LIC Pension Fund, Unit Trust of India Mutual Fund and General Insurance Corporation of India. According to the Sebi order, the two front-ran 1,220 trades including 986 in the cash segment to profit more than Rs 1.28 crore and 234 in the F&O segment to profit more than Rs 32.82 lakh.
The order stated that there are adequate additional circumstances present in the matter such as the IP address, the close relationship and financial connection between the two, particulars of trading activity during the investigation period, increased risk appetite of Savla, attempt made by Dhedia to hide his identity / digital trail and so on that further strengthens the allegation of front running against Dhedia and his sister Kajal Savla.
Probability calculations
The Sebi order calculated the probability thus:
There are 6.5 hours in a normal trading day in India, between 9:00 am and 3:30 pm. Let us consider only the time ranged that we observed Noticee No. 2’s (Savla's) trading in HUL in the 20 occasions (when both Noticee No. 2 and LIC through IDBI Securities happened to trade in HUL on the same day), i.e. between 10:30 am and 2:45 pm, or during an interval of 4 hours and 15 minutes (255 minutes).
LIC could be commencing purchase of HUL through IDBI Securities anytime during this 255-minute period. What is the probability that an investor happens to commence purchase of HUL within 3.5 minutes (Savla was observed to be buying the stock between 15 seconds and 3.5 minutes before LIC does on the various occasions) before the commencement of purchase of HUL by LIC through IDBI Securities (hereinafter referred to as ‘FR Event’)?
The order quoted the classical definition of probability: If there are ‘n’ mutually exclusive, equally likely and exhaustive cases out of which ‘m’ of them are favourable to an event A, then
Probability P(A), p = m / n
Thus, the probability of the FR Event would be p=(3.5 / 255), or around 1.37%. The order noted that this looks like a low number, but it is "certainly not improbable".
To extend this over 20 separate occasions (as was observed in Savla's trading), Narayan used Binomial Probability Distribution (see image below).
Applying it to the current case, where the probability of the FR Event happening once is 1.37 percent, the probability of the event happening more than once in 20 instances was calculated as below
It gave the the probability of the FR Event happening more than once in 20 occasions as approximately 3.04%; "again, while this is low, this is certainly not improbable".
Extending it to more than 5 occasions out of the 20, the probability is 2.2 times in 10,000,000 (10 million) – "a very, very low probability indeed".
More than 10 occasions out of the 20, the probability is now 4.88 times in 10,000,000,000,000,000 (10 Quadrillion, or 10 followed by 15 zeros). The order stated, "to put it mildly, this suggests extreme improbability".
More than 15 occasions out of the 20, the probability is now 7.3 times in 1,000,000,000,000,000,000,000,000,000 (1 Octillion, or 1 followed by 27 zeros).
As the order noted, "For all practical purposes, this probability number suggests a near-impossibility".
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