Finance — Breaking the Market

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.

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