Summary of selected posts from Breaking the Market finance blog. As with Ergodicity Economics, I don’t pretend to understand all of the underlying math or economic concepts, but some of the takeaways seem nonetheless evident.

Key Takeaways

• Investing: be weary of averages.
  • Average (market) performance can be misleading.
  • What happens on average is not the same as what happens to one investor over time.
  • The (market) average is skewed upwards by the extreme (out-)performance of the few (winner take all).
  • Over time, individual outcomes are increasingly likely to be below the (simple) average.
• Investing: minimize volatility.
  • Volatility reduces individual outcomes.
  • Diversify.
    • Play multiple games at the same time.
    • Invest in indices versus individual stocks.
    • Invest across negatively correlated asset classes.
  • Re-balance.
    • Re-start the “game”.
    • Regularly apply and update your investment weightings.

The importance of the geometric average

• Games of chance.
  • Single games or a series of games (or bets).
  • Very different odds of success.
• Single games are additive.
  • One single bet.
  • Or a group of simultaneous bets.
  • Driven by the arithmetic average.
    • (1+2+3+4+5)/5 = 3
  • Arithmetic average:
    • The value when added up the appropriate number of times = addition of individual factors.
    • 1 + 2 + 3 + 4 + 5 =15
    • 3 +3 +3 + 3 +3 = 15
• Series of games are multiplicative.
  • Repeated bets (series).
  • Driven by the geometric average.
    • (1 x 2 x 3 x 4 x 5)^1/5 = 2.6
  • Geometric average:
    • The value that when multiplied by itself the appropriate number of times = the multiplication of individual factors.
    • 1 x 2 x 3 x 4 x 5 = 120
    • 2.6 ^ 5 = 120
• Many bets we care about are multiplicative.
  • The arithmetic average becomes meaningless.
  • Arithmetic average represents an increasingly unlikely outcome as repetitions increase.
  • Given enough rounds, the average will degrade to the geometric average.

The geometric average is always less than arithmetic average. 

• Formula:
  • Simple average – (standard deviation ^ 2) / 2
  • Volatility / variation reduces outcome.
• Simple explanation:
  • Price goes up by 10% and down by 10%.
  • Arithmetic average would be (10% – 10%)  / 2 = 0%
  • You actually end up with 1.1 * 0.9 = 0.99.
  • The end result is -1%.
• Volatility “ate up” 1% of your pot.

Benefits of diversification.

• Adding “games” (stocks, coins) slows down the process of convergence towards the geometric average.
  • Allows you to play multiple games at the same time.
• The more simultaneous games, the closer you stay to the arithmetic average.
  • Re-balancing between games is important – see below.

Additive games are more likely to be profitable than multiplicative games.

• Danger of multiplicative games:
  • Tends to generate a large number of substantial losers and only a few winners.
  • Arithmetic average becomes especially deceiving, as it deviates substantially from the pattern of outcomes.
  • Danger of relying on the simple average:
    • Over many repetitions, returns trend toward the geometric average.
    • The geometric average may be negative, even when the simple average is positive.
• Steer towards additive games:
  • Create a scenario where you re-start the bet (end the previous bet).
  • Stay closer to the simple average.
  • Determine the amount you bet each round.
  • As the bet size changes, the long term odds of the game change.

This means: re-balance your portfolio.

• Buy and hold = game with many rounds = drive return toward geometric average.
  • Becomes very skewed by investing in only a handful of stocks (the winners) as time goes by.
  • The lack of balance in the portfolio reduces the portfolio’s diversification.
  • This increases the portfolio’s volatility.
    • Special case: return exceeds volatility (Buffett exception) = buy and hold.
  • The increased volatility reduces the geometric return.
• Rebalancing = ending the game = drive return toward arithmetic average.
  • Rebalancing reduces volatility drag.
  • The shorter the time period, the better the results.
  • Balance with transaction costs of re-balancing.

Geometric balancing

• Optimal strategies for random games exist, driven by:
  • Geometric averages.
  • Bet size.
  • Re-balancing across gambles.
• Practical restrictions:
  • Asset returns over any short time frame are extremely unpredictable.
  • Volatility is fairly predictable.
  • Correlation is slightly predictable.
• Mix assets based on:
  • Geometric returns.
  • Variance.
  • Correlation.

Portfolio construction

• Geometric return = average return – volatility drag.
  • Maximize return.
  • Minimize drag.
• Maximize return:
  • Stocks have highest historic return, but high variance (30%).
  • Index reduces volatility to about 16%.
• Minimize drag:
  • Negatively correlated assets.
    • Stocks.
    • Treasury bonds:
      • Treasury notes / bonds are most highly traded assets with negative correlation to stocks.
      • Long-term treasury bonds have highest yields.
    • Gold.
    • Cash.

Avoid negative geometric return.

• Game will always trend toward losses with repetition.
• Even if the game has a positive return for a single play.
• May take more or less rounds for the effect to grow.

The equity premium puzzle

• Standard general equilibrium model:
  • Equity premium less than 1% for reasonable risk aversion levels.
  • Calibrated over business cycle fluctuations.
• Sharp contrast with the historic average equity premium of about 6%.
  • Range of about 3-7%.
  • Requires either a large risk aversion coefficient.
  • Or counter-factually large variability.
• Returns typically expressed as arithmetic average.
  • Long term average about 6%.
• Geometric average is much lower.
  • Market: 6% – (0.20^2)/2 = 4% (assuming 20% standard deviation).
  • Stock: 6% – (0.33^2)/2 = 0.6% (assuming 33% standard deviation).
• Geometric return of individual stocks are closer to risk free return.
  • Smaller implied equity premium.

Analyzing historical versus future events

• Major difference between:
  • Statistics used to review prior historical events.
  • Statistics used to predict future events.
• Historical studies:
  • Analyzing a catalog of prior events.
  • Selection bias:
    • The events you study are taken out of the sample size (sampling w/o replacement).
    • Overlapping nature of streaks.
  • This changes the statistical characteristics of the remaining sample.
    • Simple example.
    • 100 coin flips: 50 heads, 50 tails.
    • Study: streak of 5 heads in a row.
    • What are the odds that next coin is also a head.
    • Not 50%.
    • Only 45 heads remaining in sample, only 95 coins remaining: 45/95 = 47%.
      • Actually odds lower are 36% due to overlapping nature of streaks.
  • An infinity long sample won’t be affected by this.
  • But we don’t live in a world of infinitely long events.