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.

- Negatively correlated assets.

**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.