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Counter-intuitive results

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Counter-intuitive results

Mathematics surprises us when our intuition is built on a faulty foundation. A loss of 50% requires a gain of 100% to recover. This violates the symmetry we expect. A portfolio with zero average return can lose money due to volatility alone. A fund that gains 12% per year on average might give you only 8% per year if you're unlucky about when you get those returns. These aren't tricks or quirks; they're baked into the mathematics of compounding.

This chapter catalogs the surprises: the asymmetry of gains and losses, the difference between arithmetic mean and geometric mean, the way volatility eats returns, the illusion created by survivorship bias, and the math behind why the investor's actual return lags the fund's reported return. We'll also explore leveraged ETF decay, why a 100% stock portfolio doesn't always beat a balanced portfolio even over 50 years, and why some seemingly-obvious investment principles turn out to be backward once you do the math.

Each of these results seems to violate intuition until you work through the mathematics. Then they become obvious—sometimes disappointingly so. But that clarity is useful. Once you understand why, you can design your finances to account for these effects instead of being blindsided by them. You'll stop arguing with the mathematics and start using it.

Averages lie (sometimes)​

An investment that returns 50% in year one and loses 30% in year two has an average return of 10%. But your actual return is lower, because you lost 30% of a larger amount. If you started with $100,000, after year one you have $150,000. After year two, you have $150,000 times 0.70 = $105,000. Your actual return is 5%, not 10%. The arithmetic mean (10%) tells you one thing; the geometric mean (5%) tells you what you actually experienced.

This matters enormously because most market data shows arithmetic means, while your portfolio experiences geometric returns. If you're comparing investments or evaluating a fund manager's track record, you need to know which mean they're quoting. A fund that claims 10% average returns might actually be delivering 6% or 7% geometric returns due to volatility. The difference is compounding, and over 30 years, it becomes life-changing.

Asymmetry and the cost of volatility​

A volatile return sequence at the same average rate will produce lower wealth than a smooth sequence. This is a feature of compounding, not a bug. Your portfolio's value follows a path, and paths matter. Two investors with the same average return but different paths end up with different wealth. This is why sequence risk is so important in retirement. The order and magnitude of returns determines your final wealth, not just the average. Understanding this reverses how many people think about risk: volatility itself is a cost that you pay, independent of whether you recover your losses.

Articles in this chapter​