Mean return, volatility — the shake callback

The shake of returns

You know mean and standard deviation from Track 1 lesson 6 — average and the width of the band around it. Now we apply them to returns. The result is finance's most-used pair of statistics.

Take a stock's daily returns over a year. Compute their mean — that's the average daily return. Compute their standard deviation — that's volatility, written σ. Two numbers describe most of what you need to know about that stock's behavior over the period.

Volatility is just standard deviation wearing a finance hat

Same math as Track 1, applied to returns instead of generic numbers. Big σ = wild stock; small σ = boring stock. The "shake" base class doing its thing.

Some rough numbers (annual volatility, ballpark):

• Treasury bills: σ ≈ 0.5% — barely shakes
• Investment-grade bonds: σ ≈ 5%
• S&P 500: σ ≈ 15-18% in normal times, 30%+ in crises
• Individual large-cap stock: σ ≈ 25-35%
• Small-cap or speculative tech: σ ≈ 40-60%
• Crypto: σ ≈ 60-100% (the casino end)

The relationship between σ and "risk" isn't perfect, but it's close enough that finance defaults to volatility as the proxy for risk. Track 9 (portfolio) refines this.

Time scaling — the √t rule

Volatility scales with the square root of time. If a stock has daily volatility σ_d, its annual volatility is roughly:

(252 because there are about 252 trading days in a year.) Why square root? Because returns over independent days add up like a random walk, and the variance of independent additions adds linearly. The standard deviation, being the square root of variance, scales by √t.

Don't memorize the proof. Just lock in: volatility scales with √t, not t. So a daily 1% σ becomes about 16% annually (since √252 ≈ 15.9). A weekly 2% σ becomes about 14% annually (√52 ≈ 7.2).

Why this matters for everything that follows

Volatility is the input to:

• The Sharpe ratio (return per unit of σ — Track 9)
• The Black-Scholes option pricing formula (where σ is the most important input — Track 8)
• The VIX index (the market's expected forward σ — Track 4)
• Value at Risk (VaR) — how much you might lose on a bad day, computed from σ
• Anything calling itself "risk-adjusted"

Burn this in: when finance says "risk," it usually means σ. When it says "risk-adjusted," it usually means "divided by σ." When it says "volatility," it means σ. Same number, three names.

The takeaway

Mean of returns = average performance. Standard deviation of returns = volatility = σ = the shake. The most common statistic pair in finance. Volatility scales with √t (a daily σ × √252 ≈ annual σ). Almost every "risk" measure is built from σ. From here on, σ shows up constantly — now you know what it actually is.