Better Betting Odds
How to Calculate Betting Odds Well
If we are presented with a single bet and we just want to consider whether or not to take it, it might be productive to just calculate the bet’s expected value - which is what people usually do. In life, we need to consider whether to take some investment or career bets which affects our bag/wallet/portfolio repeatedly, we will make these bets again and again year after year. When we just use standard expected value(arithmetic average) to decide whether or not to take some bet we miss much information. It is possible - even likely - that we will go bust on a favorable bet that has negative returns if we repeat it year over year, e.g. very high interest bonds. When we instead use the geometric average to calculate the expected returns of some bet, we take into account our probaility of going bust and compounding returns.
For example, if we choose between two bets for which we will repeat 10 times and with the same arithmetic average of 20% return on investment.
Bet #1 - 9/10 chance of 30% return on investment and a 1/10 chance of a 70% loss.
Bet #2 - Always a 20% return on investment.
Both bets have an arithmetic average of a 20% yearly return but the geometric averages show expected 10 year returns of 318% and 620% respectively, equivalent of 12% and 20% yearly returns respectively. It’s not even close but we’d loose all this critical information if we only check the arithmetic average.
These values are calculated as,
Bet #1 - (1.3 ^ 9) * .3
Bet #2 - 1.2 ^ 10
This is helpful to know so we can be more precise and confident while making decisions, but using geometric average is criticial when we face chances of negative returns like the Saint Petersburg paradox. In the Saint Petersburg paradox we are shipping cargo with an expected return of 10% per shipment across some canal, but there is a chance that pirates will seize each ship netting us a -100% loss. In this example we find it is much more profitable and effective to pay the cost to insure our ships and break up the shipments between several ships. Even though insurance will give us a percentage cost across the board, it wipes out the downside and increases our geometric returns tremendously.
References
Universa Investments on Saint Petersburg Paradox
Safe Haven: Investing for Financial Storms by Mark Spitznagel