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Probability-weighted Return

The Difference Between Probability and Confidence

Cameron Hight discusses the difference between probability and confidence in forecasting.

Probability and confidence are often confused because both are used in forecasting and are expressed as percentages ranging from 0% to 100%. However, they are distinct concepts. In our 2009 blog, “The Probability Problem”, we illustrate the difference with an example:

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 CONFIDENT about the distribution of probabilities.

Alpha Theory clients become conversant in the difference because probability and confidence are independent variables that inform the manager’s position size.

In the example below, the probabilities are associated with the potential outcomes for the security and impact the Probability-Weighted Return (PWR).

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Confidence is based on our belief that our forecasts are correct. Our clients build checklists to inform their confidence:

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Each of these independently influences the weight of the position in the portfolio. See below how the probabilities impact the PWR which lowers the position size from its max of 10% to 6.6%. Then the 45% confidence lowers the position size from 6.6% to 3.0%.

Michael Mauboussin’s “Confidence: Methods to Assess Confidence Under Uncertainty” explains in greater detail the importance of thinking differently about probability and confidence and provides various approaches for assessing both. Here are a few of the highlights:

Investing is an activity that is inherently probabilistic. Nearly all investment opportunities present a range of possible outcomes with some chance of occurring. The goal is to invest in situations where the expected value, the sum of the potential outcomes times the probability that they happen, is different than the price.

Confidence can play an important role in the investment process. For example, the price of two investment opportunities may present the same discount to expected value, but confidence in the probabilities for one may exceed those of the other. That nuance may be relevant for determining the appropriate weighting of securities within a portfolio or evaluating diversification.

We have worked with 300+ funds and 1000s of analysts over the prior two decades helping them build their own unique position sizing framework. We have seen the benefits for a team of making both probability and confidence explicit inputs of the position sizing process. The difference between probability and confidence is not common knowledge but once explained, it becomes common sense.  

Probability-weighted Return