The dangers of adding and subtracting percentages

If a stock gains 20% today and loses 20% tomorrow, are you back to where you started? Most people instantly answer yes, assuming that equal percentages cancel each other out.They don't.The real answer is that you aredown 4%. This mistake comes from treating sequential percentage changes as additive instead of multiplicative. Percentages are not fixed, absolute amounts - they arerelative multiplierstied directly to the number they modify. When a value increases by 20%, youmultiply it by 1.20. The subsequent 20% loss applies to thatnew, larger figure, taking away a bigger dollar amount than you gained in the first place. Here is the exact math: Start with $100. Add 20%: $100 × 1.20 = $120. Subtract 20%: $120 × 0.80 = $96. You can apply the same sequence of calculations to a general variableY. Start withY. Add 20%:Y× 1.20 = 1.20Y. Subtract 20%: 1.20Y× 0.80 = 0.96Y. Multiplying the two stages together gives an overall factor of1.20 × 0.80 = 0.96. This leads to a loss of 4% -regardless of the initial starting value. In finance, ignoring this property leads to serious miscalculations. It hides the asymmetry of losses: A 20% drop requires a 25% gain just to break even. A 50% drop requires a 100% gain just to break even. When working with percentages, it is good to start with the baseline value explicitly and perform the calculations one step at a time - just like I did above. This will make it easier to obtain the final answer correctly. Last week, I wrote about another common mistake involving percentages:interpreting an increase of 200%. I encourage you to read that article, too!