Compounding Concepts5 分で読めます

Negative Compounding — When Losses Grow Through Compounding

You fell 10%, so a 10% rise gets you back to even, right? Sadly, no. Compounding works in reverse on losses, so you have to rise by more than you fell just to barely return to where you started.

Why Does Recovering from -50% Require +100%?

Say you have about $740. If 50% is lost here, about $370 remains. Now, to get back to about $740, how much must you rise?

You need to earn another $370 on top of $370, so on the basis of the remaining money, +100% is required. -50% and +50% look equal in size, but the 'basis (denominator)' when you rise is already the halved $370.

Here's the key. A loss is recalculated on the basis of 'the money you have left now.' That's why the percentage you fell and the percentage you must rise again are never the same.

Required recovery-rate formula: recovery rate = loss rate ÷ (1 − loss rate). For -50%, that's 0.5 ÷ 0.5 = 1.0, i.e., +100%.

The Bigger the Loss, the Exponentially Harder the Recovery

Each time the loss grows a bit, the required recovery rate jumps far more steeply. Look at a table computed with the formula (loss rate ÷ (1 − loss rate)).

① -10% loss → about +11.1% needed ② -20% loss → +25% needed ③ -50% loss → +100% needed ④ -90% loss → +900% needed

Up to -10% the difference doesn't look big, but once you pass -50%, the recovery burden explodes. This is exactly 'negative compounding,' the phenomenon of losses growing heavier through compounding. It's the passage that shows why avoiding a big drawdown is so heavily emphasized.

The same formula and figures are confirmed in various calculators and sources (Bogleheads, Exceljet, etc.).

A Real Case History Showed: the Dot-Com Bubble

This isn't theory but something that actually happened. The U.S. Nasdaq Composite Index peaked at 5,048.62 on March 10, 2000.

After that, as the dot-com bubble burst, it fell to around 1,140 by around October 2002. That's a drawdown of roughly -77% to -78% from the peak. To recover a loss of this magnitude back to principal, the formula requires roughly +350% or more.

In fact, the Nasdaq did not surpass its 2000 peak again until April 23, 2015 — about 15 years later. A big drawdown isn't simply 'a large decline'; it's also the problem of taking a long time to recover.

The peak of 5,048.62 and the prior-peak recovery on 2015-04-23 are cross-checked against multiple sources such as Wikipedia and Goldman Sachs. The size of the drawdown is stated in the -77% to -78% range depending on the source.

Ups and Downs Create Losses Too: Volatility Drag

Losses can accumulate even without one big drop. This is called 'volatility drag' (erosion caused by volatility).

For example, say you recorded +50% one year and -50% the next. On an arithmetic-average basis it looks like 0%. But in reality it goes 100 → 150 → 75, a -25% loss. Even if the average is unchanged, the greater the swings, the smaller the compounded return you actually pocket.

So the greater an asset's swings (volatility), the wider the gap between the 'average return' and 'the money I actually earned.' This is the reason a big drawdown and excessive volatility are guarded against in long-term investing.

The principle: arithmetic mean ≥ geometric mean (actual compounding). Tom Messmore organized this in 1995 as the concept of 'volatility drain.'

よくある質問

Q. So are losses unconditionally bad and must be avoided entirely?

Loss stretches naturally accompany investing. The key isn't to make loss itself 0, but to avoid the 'very large drawdown' from which recovery is practically difficult. -10% and -70% have completely different recovery difficulties. This isn't about telling you to buy or sell a particular stock; it's about understanding the effect the size of the drawdown has on recovery.

Q. How do I calculate the required recovery rate myself?

Recovery rate = loss rate ÷ (1 − loss rate). For -30%, that's 0.3 ÷ 0.7 ≈ 0.4286, i.e., about +42.9% is needed. If you plug in the numbers yourself, you can feel how sharply this value grows as the loss rate rises.

Q. Can I verify this concept with real data?

Yes. 'The Return of Almost Everything' is a site I run personally, and it shows the maximum drawdown and recovery period of crisis periods with real data. When you see with your own eyes how much longer recovery took as the drawdown grew larger, this principle sinks in more.