# Probability Weighted Payoff

I came across an interesting example discussing the merits of probability weighted payoffs in binary scenarios. Very applicable in event driven funds. Anyway the example is below:

“Jacques had a post-announcement merger arbitrage idea. He presented it to the investment committee saying the stock traded at €7.80, the company had a takeover offer on the table for €8.00. The success or failure of the deal should be known within 6 weeks. That’s a 2.6% gain over the life of the investment, or a 25% annualized gain. Not a bad day at the office.”

“Or is it? Prior to the announcement, the target company traded at €6. This means that at €7.80, the market prices in a 90% chance the deal gets approved by regulators, the target’s shareholders and competition authorities (and various other deal terms too numerous to mention). For this to be a positive expected outcome investment, you need to believe the probability of approval is higher than 90%. So on a probability-weighted basis, it’s not that attractive.”

"Let’s put some numbers to it:

" There are two problems here:

1. The expected outcome is 0, so it doesn’t make sense to execute the trade
2. If you’re wrong, you’d lose 23% in a very short time frame. The rather vicious mathematics of this means you’d need to subsequently generate +30% returns just to break even.

The author goes on to extrapolate this to his covid positioning which is really interesting to see how the probability weighted payoffs can help guide decision making.

"Let’s look at this in terms of portfolio positioning using my own experience in 2020. In early February 2020, my portfolio was up about 5%, not bad, but nothing to write home about. At this time, I didn’t think covid was going to be a big deal in terms of economic impact. But I did think that if it became a big deal, it would have some pretty nasty consequences. The question I asked my self was not “will Coronavirus hurt the stock market” but “what probabilities and magnitudes of impact do I need to believe to stay invested”.

Let’s say the market continues to advance at its historic averages: somewhere between ~5-7% p.a. The market is already up ~1.4% this year, so by staying invested, I should expect further gains of ~1-2% over the next 3 months. What happens if covid does hit the market? Let’s say it results in a bear market, something like a 20-25% drop. What probabilities do you need to believe in for this to be a 0 expected outcome event? Let’s run the numbers.

His Blog is found here: https://www.crombiegi.com/post/right-and-wrong

Highly recommend reading it!