Math Puzzle: You Win Some, You Lose Some

Penny has lost money. It seems like growing her portfolio by 15 percent and decreasing it by 15 percent should perfectly balance out, but the math says otherwise. Imagine Penny originally invested $100. After the first month, her portfolio is worth 15 percent more: $115. To calculate a 15 percent decrease, we multiply this by 85 percent: 115 × 0.85 = 97.75, which is less than the original $100.

Intuitively, the investment increases by 15 percent of its original value and then decreases by 15 percent of its new value. That new value is larger than the original value, so the loss outweighs the gain.

In fact, solving this symbolically demonstrates that Penny will lose money regardless of her initial investment amount, the percentage by which her portfolio fluctuates or the order in which the rise and fall occur. Call the amount of Penny’s initial investment P and say we’re increasing or decreasing it by x (e.g., if we’re fluctuating by 15 percent, then x = 0.15). Then, after the first month, the portfolio is worth P × (1 + x). After the second month, this new quantity shrinks: P × (1 + x) × (1 – x). A little algebra reveals that after two months, the investment is worth P × (1 – x²). The (1 – x²) term is always less than 1 no matter what value we assign to x (unless x = 0, indicating no change). This implies that the final value of P × (1 – x²) is always smaller than the initial investment of P. Furthermore, we get the same answer if we swap the good month and bad month because the order in which you multiply terms does not affect their product: P × (1 + x) × (1 – x) = P × (1 – x) × (1 + x).