The Intuition Behind Time Diversification
A commonly cited argument in long-term investing is that time reduces risk. The logic is straightforward: over a single year, equity returns might range from -40% to +50%; over a 30-year period, the annualised return has historically fallen within a much narrower band. The idea that holding equities for longer makes them "safer" is called time diversification.
This intuition has statistical grounding. The annualised standard deviation of equity returns decreases as the holding period lengthens. For one-year US equity returns, annualised standard deviation has historically been around 15-20%. For ten-year holding periods, the annualised standard deviation of the compounded return is substantially lower — perhaps 5-7%. For 30-year periods, it narrows further. Time appears to diversify away volatility.
The concept has become the cornerstone of the life-cycle investing framework: young investors have a long time horizon and can therefore tolerate higher equity allocations.
Samuelson's Counterargument
Paul Samuelson challenged the time diversification argument in a 1963 paper, "Risk and Uncertainty: A Fallacy of Large Numbers." His insight was that while the annualised standard deviation of returns decreases with time, the absolute dollar uncertainty in terminal wealth increases.
Consider a simple example. An investor with £100,000 invested for one year faces bounded terminal wealth uncertainty. Extend the horizon to 30 years. The annualised standard deviation is much lower, but the portfolio has grown to roughly £761,000 at the expected return. A small percentage standard deviation on a portfolio of that size produces a much larger absolute pound range of outcomes than the one-year case. The investor faces more total wealth uncertainty, not less.
Samuelson's point is that time diversification is a statistical artefact: it reduces annualised percentage variance while simultaneously increasing the absolute size of what is at risk.
Reconciling the Views
Both sides capture something true. Annualised volatility does decrease with time, which matters to investors who evaluate performance as a percentage return. But absolute wealth uncertainty increases, which matters to investors who care about terminal wealth in absolute terms.
The reconciliation is that the correct measure of risk depends on the investor's situation. For an investor who needs to sustain a specific level of annual income in retirement, absolute terminal wealth uncertainty matters most — and time does not eliminate it. For an investor with flexible consumption and no specific terminal wealth target, the reduction in annualised volatility is meaningful.
Robert Bodie's 1995 paper "On the Risk of Stocks in the Long Run" made a related point using options pricing: the cost of insuring against a loss below the risk-free rate increases with time for equity portfolios. If equities genuinely became safer with time, insurance would become cheaper. The opposite is true.
The practical implication: young investors should hold high equity allocations — not because equities become safe over time, but because young investors have the capacity, flexibility, and income to withstand the volatility. The justification is the ability to bear risk, not the elimination of risk.