Your statistics background is deliberately light, so we build the two tools we need from zero. They're worth owning cold, because everything in this phase leans on them.
How do you calculate expected return?
Start with expected return, the probability-weighted average. Think of the future as a menu of scenarios, each with a probability and a payoff. The expected return is what you'd average if the future ran many times. Take Meridian Motors (synthetic), an auto maker with cyclical earnings:
| Scenario | Probability | Return | Probability × return |
|---|---|---|---|
| Boom | 0.3 | +30% | +9.0% |
| Normal | 0.5 | +10% | +5.0% |
| Recession | 0.2 | −15% | −3.0% |
| Expected return E(R) | +11.0% |
Notice what E(R) isn't. It isn't the most likely outcome, which is +10%. It may even be an outcome that cannot happen, since no scenario pays 11% on the nose. It's the long-run average of the process, and it's the number that matters for pricing, because a price is set against the whole menu and the most likely line is only one row of it.
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How do you measure variance and standard deviation?
Next comes dispersion. Two assets can share an expected return and be utterly different investments. Compare Meridian with Sundar Staples (synthetic), a soap-and-shampoo business:
| Scenario | Prob. | Meridian | Sundar |
|---|---|---|---|
| Boom | 0.3 | +30% | +16% |
| Normal | 0.5 | +10% | +10% |
| Recession | 0.2 | −15% | 0% |
| E(R) | 11.0% | 9.8% |
Similar averages, different widths. Here's how to measure width. Take each scenario's miss from the expected return and square it. Weight the squares by probability, then add them up. That's the variance. Its square root is the standard deviation, σ, the "typical miss."
Work it for Meridian, whose misses are +19, −1, −26, each measured against 11. Squared: 361, 1, 676. Weighted: 0.3×361 + 0.5×1 + 0.2×676 = 108.3 + 0.5 + 135.2 = 244.0. So the variance is 244 "squared percentage points", an unintuitive unit, and that's why you take the square root: σ = √244 ≈ 15.6%. Sundar's misses are +6.2, +0.2 and −9.8, which give a variance of 30.8 and a σ of about 5.5%. One number now captures what your eye already saw. Meridian's future is roughly three times wider.
What does standard deviation tell you, and where does it come from?
Two questions students always ask, with straight answers:
- Why square the misses instead of just averaging their size? Two reasons. First, squaring kills the sign problem. Misses above and below the mean would otherwise cancel to zero and tell you nothing. Second, it punishes big misses more than proportionally. A −26 hurts more than twenty-six times a −1, which matches how investors experience outcomes. There's also a deep practical reason you'll see in the two-asset arithmetic next. Variances add across assets in a clean way that other dispersion measures don't, and that additivity is the mathematical hinge of diversification.
- What does σ = 15.6% actually let me say? If returns were roughly bell-curve shaped, about two-thirds of years would land within ±1σ of the mean, which here means −4.6% to +26.6%, and about 95% within ±2σ. Use that as a ruler, never a guarantee. Real market returns have fat tails, and extreme moves happen far more often than the bell curve permits. On 19 October 1987 the US market fell over 20% in one day. A bell curve calibrated to daily σ calls that essentially impossible, and it happened anyway. σ is a fair summary of ordinary weather and a poor guide to storms. Phase 9 picks this up as a risk-management matter.
So where do the numbers come from in practice? Scenario tables are for teaching and for structured thinking. Practitioners mostly estimate σ from history, and the convention is five years of monthly returns (60 observations), with monthly σ scaled to annual by multiplying by √12. That scaling assumes months are independent. It's standard, and it's imperfect. Broad equity indices run at annual σ of roughly 15–25% in normal regimes. Both the NIFTY 50 and the S&P 500 fall in that band, with India typically a few points higher. Treat those ranges as illustrative. Your screener or index factsheet has the current figures. Single stocks run far wider, 25–50%+, for a reason the diversification argument makes precise.
Arithmetic vs geometric average return
Now a bridge back to the bootcamp. You already know the arithmetic average of returns overstates what you compound: −50% then +100% averages +25% but compounds to 0%. Dispersion drives the gap, and there's a beautifully simple approximation: geometric ≈ arithmetic − σ²/2. For a market with σ ≈ 20%, the drag is about 0.04/2 = 2 percentage points. File that number away. When you reach the historical equity premium, it explains, almost point for point, why that premium comes in two sizes.