Putting it all together

The final episode of the Beginner series connects time, risk and probability into one way of thinking about uncertainty.

Play with it

Monte Carlo lab. Simulate thousands of futures for a portfolio and watch the distribution of outcomes form.

Open full screen

Starting amount$10,000

Added each month$0

Expected annual return7.0%

Volatility18%

Broad stock market: about 15–20%. A single stock: often 30% or more.

Years20

Simulated futures

Median outcome$29.4K

Expected value (mean)$41.1K

Bad case (5th percentile)$6.8K

Chance of ending below what you put in9.9%

Median Middle 50% Middle 90% Total put in

Where the 1,000 futures end up

The future here is a cloud of possibilities, not one line. Across 1,000 simulated futures, the middle 90% end between $6,836 and $117,505, and 9.9% finish below the $10,000 you put in. The mean ($41,073) sits above the median ($29,370) because a few runaway winners pull the average up: the typical future is worse than the average one.

Try this

Run 100 futures, then 5,000. The individual paths stay messy, but does the shape of the distribution settle down?
Raise volatility while keeping the return fixed. Watch the median fall away from the mean.
Switch between 5 and 30 years. Time stretches outcomes apart, and risk decides how far.
Press run new futures a few times. Does any single path ever land exactly on the expected value?

The math

S_{t+\Delta t} = S_t \, e^{\left(\mu - \frac{1}{2}\sigma^2\right)\Delta t \;+\; \sigma\sqrt{\Delta t}\,Z}

$S_t$: portfolio value at time t
$\mu$: expected annual return
$\sigma$: annual volatility
$\Delta t$: one month, as a fraction of a year
$Z$: a fresh random draw from the standard normal, every step

Quick check

Two questions to see if it clicked. Answers explain themselves.

In the Money Loop, what does probability determine?

What happens to uncertainty as the time horizon grows?