# How to lose your edge to volatility

## When you buy & hold, volatility is not your friend. It slows down compound growth. We illustrate this for a simple coin toss and simulation through geometric Brownian motion. Risk management through position sizing could turn a negative compounded growth rate positive.

## 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:

- Would you play this game once?
- 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.

Does it follow that we should be willing to play many rounds?

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

What we find is that the average (blue) and median (red) diverge:

- Average wealth increases the more rounds are played
- Median wealth trends down

In other words, average wealth increases but **the average player gets poorer**. Wealth becomes increasingly unequally distributed with each round.

If we dig deeper into the range of outcomes, we find that they are highly skewed:

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

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

The key mechanic is that wealth is reinvested each round. This makes returns path-dependent and sensitive to 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 'experienced' by an individual player over time. This is the natural state of financial markets. When you hold an asset, you are in effect, continuously reinvesting.

If we were to play the coin toss with a constant $100 investment per round, we would remove path dependency, and render the game ergodic. Growth would become additive. The average across players or for a player over multiple rounds would become the same.

We could also position size as a fraction of wealth. 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 shrunk visibly
- The average player can now expect to win

In rudimentary form, this 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:

dS/S = (μ - σ2/2)t + σ√t.

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 expected 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.**

g ≈ μ - σ2/2

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.

σ^2 = 0.5 x (0.5 - 0.05)^2 + 0.5 x (-0.4 - 0.05)^2 = 0.2025

g ≈ 0.05 – 0.10 ≈ -0.05.

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/

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