Why the typical price path loses

The Monte Carlo panel on the home page draws 64 price paths a year long and stacks where they end into a histogram. Watch it for a few runs and something looks off: the model is set to grow 6% a year, yet about half of the paths end below where they started. Nothing is broken. That is what compounding does to a noisy price.

The model

Geometric Brownian motion says the relative change of a price $S$ over a short time $dt$ is a constant drift plus noise:

Here $\mu$ is the drift, $\sigma$ the volatility and $dW$ the increment of a Brownian motion, a normal draw with variance $dt$. Applying Itô’s lemma to $\ln S$ gives a closed-form solution,

so $\ln S_T$ is normal and $S_T$ is lognormal. The extra $-\tfrac{1}{2}\sigma^2$ is the whole story.

Mean and median part ways

Take the expectation and the noise term contributes $e^{\sigma^2 T/2}$, which exactly cancels the correction:

The mean grows at $\mu$; the typical path grows at $\mu - \sigma^2/2$. The gap, $\sigma^2/2$ per year, is often called volatility drag. A few lucky paths that end far above the rest pull the mean up, while most paths sit below it. The chance of ending below the starting price is

with $\Phi$ the standard normal distribution function.

The figure starts with the home page’s settings, $\mu = 6\%$ and $\sigma = 38\%$. The drag is $7.2\%$, larger than the drift, so the expected price after a year is about $106$ while the median is about $98.8$, and roughly $51\%$ of paths end below $100$. Push $\sigma$ to $80\%$ with $\mu = 10\%$ and the median falls to about $80$ while the mean rises to $110.5$; about $61\%$ of paths now lose.

Simulating it

Because the solution is exact, there is no need for a small-step integrator: draw normal increments, add the drift in log space and exponentiate.

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import numpy as np
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def gbm_paths(s0, mu, sigma, years=1.0, steps=252, n=400, seed=7):
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    rng = np.random.default_rng(seed)
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    dt = years / steps
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    shocks = rng.standard_normal((n, steps))
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    log_steps = (mu - 0.5 * sigma**2) * dt + sigma * np.sqrt(dt) * shocks
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    log_paths = np.cumsum(log_steps, axis=1)
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    return s0 * np.exp(np.hstack([np.zeros((n, 1)), log_paths]))
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paths = gbm_paths(100, mu=0.06, sigma=0.38, n=100_000)
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final = paths[:, -1]
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print(f"mean {final.mean():.1f}  median {np.median(final):.1f}  "
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      f"below start {(final < 100).mean():.1%}")

With a large n the printout lands on the theory: mean near $106.2$, median near $98.8$, a little over half the paths below the start.

Why it matters

Volatility drag is why a portfolio’s average return and its compounded return differ, why leveraged products decay in sideways markets, and why “expected value” can describe almost nobody’s actual outcome. The model is deliberately simple; real prices have fat tails and changing volatility. The gap between mean and median survives those refinements, and it only gets wider as volatility rises.