# CHAPTER 4Time Diversification

## FALLACY: TIME DIVERSIFIES RISK

Most investors are willing to allocate a larger fraction of their portfolio to risky assets if they have many years to invest than they would, given a short investment horizon, because they believe time diversifies risk. In his landmark article entitled “Risk and Uncertainty: A Fallacy of Large Numbers,”^{1} Paul A. Samuelson famously showed that investors should not increase their exposure to risky assets as their investment horizon expands.

## SAMUELSON'S BET

Samuelson was inspired by a conversation he had with an MIT colleague. He offered his colleague a chance to win $200 or lose $100 with better than even odds. His colleague rejected the bet, arguing that he could not afford to lose $100. But his colleague put forth a counter proposal in which they would enter into 100 such bets. He reasoned that the law of large numbers would ensure his success. Investors use the same logic to support the conventional wisdom that it is safer to invest in risky assets over long horizons than short horizons. The conventional wisdom follows from the observation that over long horizons, above average returns tend to offset below average returns. In his article Samuelson showed that this logic is a misuse of the law of large numbers.

## TIME, VOLATILITY, AND PROBABILITY OF LOSS

The following analysis might persuade you that time does indeed diversify risk. Consider an investment that has an annualized continuous return of 10 percent and ...