狮山巡礼
投资杂谈
The Flaw in High-Stakes Investment
Imagine a dice bet: roll a six, win 100x your net worth; any other number, lose everything. As a wealthy individual, my ability to pay is not an issue.
A trained investor might take this bet, as its expected return is nearly 16 times the initial wealth. But intuition warns against it. Your current comfort, a home and stability, would be instantly gone if you lose. The gain might not proportionally increase your happiness, but the loss would be catastrophic.
This highlights a common flaw in risk assessment. Investors often use a mental decision tree to calculate a stock's expected value—a probability-weighted average of all potential outcomes. The problem? A company’s future isn’t a simultaneous exploration of every branch. It’s a time-sequenced journey down a single path. This makes us blind to excessive risk.
The Two Averages: Ensemble vs. Time
Daniel Bernoulli (1738) first tackled this problem, arguing that math alone is insufficient. He introduced utility, the subjective value of wealth. For a poor person, an extra dollar has high utility. For a rich person, it has little. He concluded you shouldn't take the bet because the expected utility of your wealth is negative infinity.
John Larry Kelly (1956) offered a more robust view, focusing on the irreversibility of time. He distinguished between two averages:
?* The Ensemble-Average: The average outcome across a hypothetical, infinite number of identical scenarios happening simultaneously. In our bet, this average is positive because it includes a few lucky winners, masking the fact that most players would lose everything.
?* The Time-Average: The average outcome over a series of sequential events. Your ability to play tomorrow depends on what happens today. You can't simultaneously play all outcomes.
The key insight is that these two averages are not the same; the situation is non-ergodic. While the ensemble-average return is positive, the time-averaged growth rate is negative infinity. A single loss ends the game. As the saying goes, "a single misstep can ruin a lifetime."
Practical Consequences
We often mistakenly use the ensemble average for our single investment journey. This is only appropriate if you can bet small amounts across many independent outcomes.
The time perspective provides an objective upper limit for prudent risk-taking. For example, in a modified game where you can bet a percentage of your wealth, the time perspective would advise risking only a small portion (around 16%) to achieve maximum growth. Risking more would actually decrease your returns.
Much of modern financial theory ignores the effect of time, substituting it with a subjective "risk preference." This leads to a dangerous system where incentives can promote excessive risk-taking, often leading to market crashes. The time-based approach provides a crucial, objective safeguard against this.
