Three tools.
One wallet.

Each coloured line is one possible future for your portfolio. Some go up a lot. Some crash. Most land somewhere in between. The bright cyan line down the middle is the median — exactly half of all simulated futures end above it, half below.

The dashed white line is your starting value. Any path above it means you made money. Any path below means you lost money. Notice how the paths spread out over time — the further into the future, the wider the range of outcomes. That's uncertainty compounding.

Try cranking up volatility and watch the paths explode in every direction. Set speed to Slow to follow a single path as it unfolds.

Each day, the portfolio moves by a random amount drawn from a normal distribution. The formula is:

S(t+1) = S(t) × exp((μ − ½σ²) + σ·Z)

Where μ is the daily return, σ is daily volatility, and Z is random. The − ½σ² term is a mathematical correction (Itô's lemma) that keeps the expected value correct. Run this 252 times for a year, 120 times for 120 different futures, and you get what you see on the left.

The price line changes colour depending on which regime the model thinks the market is in. Green = bull market (prices trending upward, calm). Red = bear market (prices trending downward). Purple = high volatility (prices jumping around without a clear direction).

The coloured bands behind the price line make it even clearer — each band covers a period when the market was in that state. Notice how regimes last for weeks or months at a time, and then shift abruptly. Real markets behave exactly like this.

Most investment strategies assume markets behave the same way all the time. They don't. A strategy that works in a bull market can destroy you in a volatile one. Regime detection is how professional quant funds decide when to be aggressive and when to step back. The technique used here — a Hidden Markov Model — is the same type of algorithm used in speech recognition to detect which word is being spoken.

Each dot is a different way to split $10,000 between five assets — stocks, bonds, gold. The position of the dot tells you two things: how far right it is = how risky that portfolio is. How high it is = how much return it's expected to earn.

Brighter purple dots have a higher Sharpe Ratio — they're earning more return per unit of risk they're taking on. Dimmer dots are worse deals. The glowing purple dot is the single best portfolio found — the Maximum Sharpe Portfolio. The curved edge they all cluster against is the Efficient Frontier.

There are infinite ways to split your money between assets. Most of them are suboptimal — you're either taking more risk than you need to for the return you're getting, or leaving return on the table. The Efficient Frontier shows you the boundary of what's actually achievable. Any portfolio sitting below this curve is a wasted opportunity. This is the foundation of modern portfolio theory, developed by Harry Markowitz in 1952. He won the Nobel Prize for it.