How to lose your edge to volatility

Let's play a game

We toss a balanced coin:

• If it comes up heads, your wealth increases by 50%
• If it comes up tails, your wealth decreases by 40%

Ask yourself:

1. Would you play this game once?
2. Would you play it multiple rounds?

The answer to the first question should be yes.

Each toss has positive expected value:

50% x 1.5 + 50% x 0.6 = 1.05.

Should we therefore play multiple rounds?

Let's examine this game assuming you bet your entire wealth each round. We simulate 1000 players who each toss the coin 50 times. Each player starts with $100.

Average (blue) and median (red) diverge:

• Average wealth increases with more rounds played
• Median wealth decreases instead

This reflects growing wealth inequality. The average player does not get the average wealth growth, they get the median, and get poorer the more they play.

After 50 rounds, the wealth outcomes are highly skewed:

• The top 1% of players obtain a return of >1000x
• But approx. 75% of players end up losing money
• Most players are down bad, ending up with <$20

This may appear paradoxical. Each coin toss has positive expected value. Every player in this game should have edge. Why do most players blow up?

The key mechanic is that wealth is reinvested each round.

Reinvestment makes returns path-dependent, and therefore sensitive to volatility realized over the path.

The problem with this game, is that the edge is too small relative to its innate volatility. The more rounds are played, the more likely one encounters a string of losses that cannot be recovered from.

To gain more intuition into this, let's examine the different states of the game after just two rounds. There are four outcomes:

• 1/4 chance of 1.5 x 1.5 = 2.25
• 1/4 chance of 1.5 x 0.6 = 0.90
• 1/4 chance of 0.6 x 1.5 = 0.90
• 1/4 chance of 0.6 x 0.6 = 0.36

The average has gone up: (2.25 + 2 x 0.9 + 0.36) / 4 = 1.1025. It scales at 1.05^2.

The median however has gone down: 1.5 x 0.6 = 0.90. It decreased by ~0.95^2.

This coin toss experiment is an example of a non-ergodic system. The average across players at one point in time is different than the average over time experienced by an individual player.

This is the natural state of financial markets. When you hold an asset, you are in effect, continuously reinvesting, while constantly flipping a proverbial coin.

If we were to play the coin toss with a constant $100 investment per round, we would remove path dependency. The game would enter a state of ergodicity. The average across players and the average over multiple rounds would become the same. Wealth growth from buy & hold would become additive.

We could also position size as a fraction of wealth and do what is referred to as Kelly sizing.

Here we run the simulation again if each player reinvests 25% of their wealth each round.

The result is that the average player’s outcome improves dramatically:

• The median now slopes upwards
• The left tail has visibly shrunk
• The 'average' player can now expect to win

In rudimentary form, this illustrates the idea behind Kelly sizing. Given an expected return and a volatility parameter, there is an optimal fractional betting size to maximize the compounded growth rate.

Volatility drag

To simulate financial markets, Geometric Brownian Motion (‘GBM’) is widely used, including in the Black & Scholes model:

GBM assumes normally distributed return at each time step. The price in each step is a return multiplied with the previous price. Prices from a GBM process are lognormally distributed. GBM prices therefore are path-dependent similar to our coin toss game.

Let's visualize what happens if, for a given return μ, we simulate different levels for σ.

What we find is that for higher volatility:

• The terminal price is higher on average
• The distribution becomes right-skewed
• The median shifts left however

The fundamental trade-off is that volatility increases the possibility of an extreme outcome at the expense of a higher likelihood of losing money.

In the Black & Scholes model we see this trade-off too. If we increase volatility of the underlying, the delta of a call N(d1) will go up, but the probability of expiring in-the-money N(d2) will go down. At the extreme of infinite volatility, a call will have a delta of one, yet an infinitesimally small chance of expiring in-the-money.

Could one make the case for preferring the higher volatility distribution because of the bigger right tail?

When a process is path-dependent, we have to look at compounded growth. In the presence of volatility, the compounded return is always lower than the arithmetic return. Phrased mathematically, if σ > 0, the geometric rate of return g is always lower than the arithmetic return μ. This phenomenon is also known as volatility drag.

It turns out that we can use the median as an estimate for geometric growth. As shown by our GBM simulation, higher volatility results in lower compounded growth.

If you buy & hold, volatility is not your friend.

We can take the formula for variance drag and apply it to our non-ergodic coin toss.

We could have quickly determined our coin toss was a bad idea if played over multiple rounds. The volatility of the game was too high relative to the expected return per round. We should only be willing to play if we can manage volatility by position sizing.

References

https://ergodicityeconomics.com/2023/07/28/the-infamous-coin-toss/

https://taylorpearson.me/ergodicity/