Empirical Properties of Stock Returns

By the end of this lesson, you should be able to:

• 1Interpret historical risk-return data and its limitations as a forecast.
• 2Explain why individual stocks are riskier than diversified indexes.
• 3Recognize same-period co-movement (beta) versus lagged predictability.
• 4Describe volatility clustering and regime-dependent volatility.
• 5Identify fat tails and why the normal model is only approximate.
• 6Evaluate historical anomalies critically — they are not guaranteed free money.
• 7Preview how CAPM and APT organize these empirical questions.

The table below summarizes historical monthly statistics for several asset categories over 1946–2001. It shows what actually happened — not what is expected to happen going forward.

Historical monthly statistics, 1946–2001 sample. NOT current expected returns. VW = value-weighted; EW = equal-weighted. Source: MIT 15.401, Lecture 12.

Plotting mean return against standard deviation reveals the historical pattern:

Higher-volatility categories generally had higher average returns historically. This is suggestive of a risk-return tradeoff, but it is not proof that every volatile security will outperform. It is a long-sample average, not a guarantee.

A single company carries both systematic and idiosyncratic risk. A diversified index averages across many companies, washing out most idiosyncratic risk and leaving mostly the systematic component.

The single stock's volatility is more than double the index's. Higher SD means returns commonly move by more percentage points away from the average — bigger surprises, both up and down. This is the practical cost of failing to diversify.

When you plot a stock's return in each period against the same period's market return, an upward-sloping cloud appears. This is the visual signature of beta from Lesson 5.4.

The slope of the best-fit line tells you the stock's beta. The scatter around that line — the vertical noise — is the idiosyncratic component that diversification reduces.

Co-movement and predictability are different things:

Just because stocks move together within a period does not mean you can predict next period's return from this period's. Same-period correlation reflects shared exposure to contemporaneous shocks — not a forecasting rule.

A positive expected return does not mean a smooth upward path. Stock prices can look like a random walk with drift — a positive average trend buried under large period-to-period noise.

Five periods of returns: +18%, −9%, +4%, +22%, −13%. Despite the volatility, the cumulative wealth path trends upward because the average return is positive. But no single period's return is predictable from the previous one.

Volatility is not constant. It moves through regimes — calm periods, rising stress, crisis spikes, and gradual normalization. This is called volatility clustering.

The common annualization rule ${\sigma }_{\text{annual}}={\sigma }_{\text{monthly}}\times \sqrt{12}$ assumes variance is stable through time. That is a useful approximation, but it can understate risk during turbulent regimes and overstate it during calm ones.

If returns were perfectly normal, about 68% of outcomes would fall within one standard deviation of the mean and 95% within two. Real stock returns deviate from this in important ways:

• Fat tails — more extreme outcomes than the normal model predicts.
• Negative skew — crashes are larger than equally large rallies.
• Changing volatility — the distribution itself shifts across regimes.
• Crash risk — occasional very large negative returns.

Researchers have documented several historical anomalies — patterns where certain types of stocks tended to outperform what simple models predict. Each has multiple candidate explanations.

A historical pattern does not guarantee future profits. Real-world investing involves frictions and risks that erode or eliminate apparent edges:

• Transaction costs and bid-ask spreads eat into gross returns.
• Taxes reduce net returns, especially for short holding periods.
• Short-sale constraints prevent exploiting negative anomalies.
• Data mining — testing many strategies on the same data inflates false positives.
• Publication bias — successful strategies get published; failures do not.
• Crowding — once an anomaly is known, capital flows in and shrinks it.
• Structural change — the economy and markets evolve, eroding old patterns.
• Hidden risk — the strategy may work until a tail event reveals its exposure.

The empirical patterns above — co-movement, risk-return tradeoffs, anomalies — raise a theoretical question: how should expected return depend on risk? Two foundational models organize this question. We introduce them conceptually here; a later module evaluates them formally.

This lesson presents the empirical questions: what do returns look like, and what patterns challenge simple models? A later module evaluates whether CAPM, APT, or multifactor models answer those questions adequately.