The Probability Problem

“The fundamental law of investing is the uncertainty of the future.”
– Peter Bernstein, famed investor

I am offered two bets. In bet number one, I am paid $150 for every heads and pay $100 for every tails. My risk-adjusted return is 25%. In bet number two, I’m presented with a bag of poker chips that are only black or white. I’m paid $150 for each white chip I pull out and I have to pay $100 for every black chip I pull out. I don’t know the distribution of colors, so my probability assumption would be 50/50. Drawing poker chips also has a 25% risk-adjusted return. Would I be equally likely to make both bets? No, I prefer the coin-flip bet because I am more certain about the distribution of probabilities.

To try and balance this issue, let’s assume that we could, with reasonable certainty say the range with which our poker chip probabilities would fall. In this example we’ll assume that white chips are somewhere between 30% and 70% of the contents of the bag. This widened distribution takes into account my uncertainty regarding my probabilities. Unfortunately, if I plot out every payout between 30% and 70% probability of success, I get an average of 25%. I’m back at square one.
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What about betting systems that constrain loss? If I use Optimal-F (Kelly) suggested bet size, I get 17% bet for the coin-flip, which is the same as the average of all of the Optimal-F bets between 30% and 70% probability. Alpha Theory optimal position sizes suffer the same issue with a position size equal for both coin-flips and poker chips.
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Here is my simple solution until I understand a better Bayesian solution. I have a somewhat arbitrary Analysis Confidence rating. Let’s name them High, Medium, and Low. The coin-flip is definitely “High Confidence” because I am certain about my coin-flip probabilities. The poker chips are “Low Confidence” because I know nothing about their true distribution. But my knowledge about the poker chips is not static. The probabilities are epistemic because, as I draw more poker chips, my knowledge of the distribution of chips will improve. I will adjust my probabilities as I draw chips and change my Analysis Confidence from Low, to Medium, and eventually to High when I have a better grasp on the distribution of chips in the bag. To account for uncertainty, I’m going to cut my bets. If I have Low Analysis Confidence, I cut my suggested bet in half, if I have Medium I cut it by 25%, if it is High, I don’t cut my bet at all. This is certainly imperfect, but it does create the effect we are shooting for, less exposure when we have less certainty in our assumptions.
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This, of course, applies to equity investing. You may have high certainty in your probabilities for one investment and only low certainty in another. They both may have the same Risk-Adjusted Return, but you are not willing to invest in them equally. Use the same Analysis Confidence constraint to adjust position size and apply a heuristic-based cut since probability theory does not have a better answer. Alpha Theory provides an Analysis Confidence setting for precisely this purpose to better refine position sizes beyond Risk-Adjusted Return.