Arbitrage odds

One of the ways I've been evaluating going-private and other arbitrage opportunities is to look at the market's implied probability of success. Like gambling situations and unlike most investment situations, arbitrage is essentially a binary, win-loss event. That makes it relatively easy to apply the Kelly Criterion.

To show how it works, let's take my first going-private transaction, Major Automotive. On October 14, 2005, the company announced plans to implement a reverse split that would pay $1.90 to shareholders of less then 1000 shares. On March 29, 2006, I bought 999 shares at $1.75 a share. After the $19.95 commission charged by my broker at the time, I hoped to make $129.90 when the reverse split was resolved. I guess that there was at least a 95% chance the split would succeed from that point since insiders owned 49.4% of the company. I don't like the idea of assigning higher odds because of Murphy's Law.

If the split failed to happen, what would my position have been worth? The most pessimistic choice would be to assign no value. It isn't unreasonable either, since I'd done basically zero research into the underlying value of the company. In that case, I'd have lost the entire $1,768.20 that the position had cost. The expected payout would be $129.90*0.95 + (0-$1,768.20)*(1-0.95) = $34.995, which isn't half bad for a zero-research investment. A slightly more realistic value would be the price of the shares just before the reverse split was announced. On October 13, 2005, Major Automotive was priced at $1.40 a share. Using that price as a floor, the expected payout was $103.93. If you are following along don't forget to include an extra commission as an added cost of failure.

As it turns out, you can calculate the odds the market is assigning to an arbitrage situation with a little bit of math. This is the equation for expected payout:

S*p + F*(1-p) = E

where:
S = price if split succeeds
F = price if split fails
p = split probablity
E = expected price

If the market is rational, you'd expect the market price to match the expected price. I used the Mathomatic Computer Algebra System to solve for p:

$ mathomatic
Mathomatic version 12.6.10 (www.mathomatic.org)
Copyright (C) 1987-2006 George Gesslein II.
50 equation spaces available, 960KB per equation space.

1-> S*p + F*(1-p) = E

#1: (S*p) + (F*(1 - p)) = E

1-> solve p

        (E - F)
#1: p = -------
        (S - F)

Now if you plug in the values for Major Automotive when I bought it, you get a 70% probability that the split would happen.

Since my probability estimate (95%) was much greater than the market's, my expected payout was correspondingly higher. Given enough of these situations profits would be astronomical for anyone who could identify them. According to the Kelly Criterion, the optimum allocation for even the pessimistic case is 27%. (This can be calculated by taking the expected payout and dividing by the maximum payout or $34.99/$129.90. The more optimistic case calls for a whopping 80% allocation.) Unfortunately, these situations are rare. The only reason reverse splits are so wildly mis-priced is that arbitrage positions are limited to a small number of shares. Unless you have a portfolio of only a few thousand dollars, it's impossible to allocate the optimal amount even under the pessimistic case.