Lesson 5 of 7
The gain a fall requires
A 37 percent fall needs a 59 percent gain to get back. Work the arithmetic, then watch it happen in two real sequences of index returns.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
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In one minute
- Recovering a fall of x percent requires a gain of x divided by one minus x, which grows much faster than the fall.
- At a compounded 10.5 percent a year, the 2008 fall needed about four and a half years of average returns to undo.
- Two real sequences from a published table each took four calendar years, which is what an average conceals.
A fall and its recovery are not symmetric, and the asymmetry is arithmetic rather than psychology. Losing half your money requires doubling what is left to get back to where you started.
The gain required to recover a fall
- After a 10 percent fall11.1%
- After a 20 percent fall25.0%
- After a 37 percent fall58.7%
- After a 50 percent fall100.0%
- After a 70 percent fall233.3%
The formula, and why it bites
A 20 percent fall leaves 80, and getting from 80 back to 100 is a gain of 25 percent. A 50 percent fall leaves 50, and getting back is a gain of 100 percent. The denominator is what does the damage: as the fall grows, the base you are compounding from shrinks.
Two real sequences
What the average rate says
Turning a required gain into years
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Take the required gain: 58.7 percent, so the index must multiply by 1.587.
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Take a rate. The same table gives a compounded annual gain of 10.5 percent for 1965 to 2025.
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Solve for the years: the logarithm of 1.587 divided by the logarithm of 1.105 is about 4.6.
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Compare with what happened: four calendar years. Close, and only because the years that followed happened to be good ones.
What the arithmetic hides
Two falls of the same size are the same on this page and are not the same to live through. The 2000 to 2002 episode arrived as three consecutive losing years, each one a fresh decision about whether to keep going. The 2008 fall arrived in one year, and the following year returned 26.5 percent.
The formula treats them identically. A holder does not, and the difference is where most of the damage is actually done: a slow decline gives many more opportunities to sell at the bottom than a fast one does.
That is the reason the arithmetic is worth doing in advance rather than during. A number written down before a fall, saying what a 40 percent decline in a holding would require and how long it might take, is a different object from the same number computed in the third losing year.
The two things this changes
The first is position sizing. A holding that can fall 70 percent needs a 233 percent gain to recover, and no plausible thesis promises that. The arithmetic is the reason a position's size is decided by what it costs when wrong rather than by how attractive it looks when right.
The second is leverage. Borrowing amplifies the fall and therefore amplifies the required gain, and it does so exactly when the borrower is least able to wait for it. A 37 percent fall in an account with no debt requires patience. The same fall in an account with borrowing can end the position before the recovery arrives, which is the subject of the leverage lesson in the sibling course.
Check yourself
What gain does a 50 percent fall require to recover?
- 50 percent
- 100 percent
- 75 percent
Show the answer
Answer: B.
Half of 100 is 50, and getting from 50 back to 100 is a doubling. The required gain is the fall divided by one minus the fall.How long did the index in the table take to pass its 2007 level after the 2008 fall?
- One year
- Four calendar years, during 2012
- Ten years
Show the answer
Answer: B.
Returns of 26.5, 15.1, 2.1 and 16.0 percent in 2009 to 2012 compound to more than the 58.7 percent required, and the level was passed during 2012.Why does leverage make this arithmetic worse?
- It lowers the required gain because the position is smaller
- It has no effect on the arithmetic
- It raises the required gain and can end the position before the recovery arrives
Show the answer
Answer: C.
Borrowing amplifies the fall, which amplifies the gain needed, and a forced sale converts a recoverable fall into a permanent loss.Sources
- Berkshire Hathaway, 2025 annual report, table "Berkshire's Performance vs. the S&P 500": annual percentage change in the S&P 500 with dividends included, for 2000 to 2012, and the compounded annual gain of 10.5 percent for 1965 to 2025.
- The cumulative falls, required gains and recovery dates are computed for this lesson from the annual figures that table prints.
What this leaves out
- The index figures are annual, so a recovery "during" a year is not a specific date, and the intra-year low was deeper than any annual figure shows.
- The 10.5 percent compounded rate is a historical figure from the table, used here as an arithmetic input rather than as an expectation.
- An index is not a portfolio. An individual holding can fall further and may never recover at all.
- Nothing here is advice about any security or allocation.