The Mathematics of Rolling Returns: Eliminating Start-Date Bias, Volatility Dampening, and True Capital Compounding
Standard mutual fund factsheets and portfolio performance presentations almost universally present point-to-point trailing returns (e.g. 1-year, 3-year, 5-year CAGR). Below is a quantitative dissection of why point-to-point metrics are inherently misleading, how rolling window distributions uncover true probability densities, and the mathematical mechanics of time-diversification.
1. The Fallacy of Point-to-Point Trailing Returns
Point-to-point Compound Annual Growth Rate (CAGR) measures the geometric rate of return from a single fixed calendar date t0 to an arbitrary terminal date tn. This creates severe endpoint dependency: if the initial date coincided with a market bottom or the terminal date coincided with an equity bubble peak, the reported CAGR will be drastically inflated.
Conversely, an exceptional fund whose terminal measurement falls immediately after a sudden 20% macroeconomic correction will appear artificially impaired. Rolling returns resolve this structural defect by sliding an identical investment window across every consecutive monthly period in history.
Pt = Asset Net Asset Value (NAV) or Index price at month t.
Pt − 12k = Asset NAV exactly k years (12k months) prior to month t.
k = Rolling horizon length in years (e.g. 3, 5, 7, 10).
2. Volatility Decay and the Law of Large Numbers
In short-term rolling horizons (k = 1 to k = 3), asset return distributions exhibit fat tails, high skewness, and extreme dispersion. As the investment tenure expands toward 7 to 10 years, the Central Limit Theorem and mean-reverting properties of economic production cause return distributions to converge tightly around the long-term earnings growth rate.
| Rolling Horizon | Typical Return Dispersion | Negative Window Probability | Investor Takeaway |
|---|---|---|---|
| 1-Year Rolling | -38% to +65% | 24% – 28% | Speculative noise dominates fundamentals |
| 3-Year Rolling | -8% to +32% | 10% – 14% | Market cycle transitions begin filtering out |
| 7-Year Rolling | +4% to +21% | < 1.5% | Dispersion narrows substantially, though negative windows still occur |
| 10-Year Rolling | +7% to +18% | 0.0% | Outcome dominated by earnings compounding rather than entry timing |