Losses are measured on the bigger base and recoveries on the smaller one, which punishes the seller and quietly rewards the saver who keeps buying.
More than you lost: the gain needed is 1 ÷ (1 − loss) − 1. After a 30 per cent fall, 100,000 is 70,000 and has to rise 42.86 per cent to get back, which takes 5.27 years at an illustrative 7 per cent a year. A 50 per cent loss needs a 100 per cent gain. The arithmetic is unforgiving for someone who sells, and kinder than it looks for someone who keeps buying.
Worked in full in Stock Market Investing for Beginners by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The asymmetry comes from the base. A fall is measured on the old, larger value and the recovery on the new, smaller one. Thirty thousand lost from 100,000 is 30 per cent; thirty thousand regained on 70,000 is 42.86 per cent. The same pair of percentages in the other order does not cancel either: a fall of 30 per cent followed by a rise of 30 per cent leaves you 9 per cent down.
| Input | Value |
|---|---|
| Portfolio before the fall | 100,000 |
| Fall | 30% |
| Return during the recovery, yearly | 7% |
| Monthly contribution (saver case) | 500 |
| Cash rate while out of the market (seller case) | 2% |
Gain needed = 1 ÷ (1 − loss) − 1
Years to recover = ln(1 ÷ (1 − loss)) ÷ ln(1 + return)
In Excel, with the loss in B1 and the return in B2: =1/(1-B1)-1 and =NPER(B2,0,-(1-B1),1).
Step 1. 100,000 × (1 − 0.30) = 70,000.
Step 2. 1 ÷ 0.70 − 1 = 42.86 per cent: 30,000 needed on a base of 70,000.
Step 3. ln(1 ÷ 0.70) ÷ ln(1.07) = 5.27 years, about 63 months, at a steady 7 per cent. Real recoveries are rarely steady, and some have been much faster or much slower; the formula only tells you what a given rate implies.
| Loss | Gain needed | Years at 4% | Years at 7% | Years at 10% |
|---|---|---|---|---|
| 10% | 11.11% | 2.69 | 1.56 | 1.11 |
| 20% | 25.00% | 5.69 | 3.30 | 2.34 |
| 30% | 42.86% | 9.09 | 5.27 | 3.74 |
| 40% | 66.67% | 13.02 | 7.55 | 5.36 |
| 50% | 100.00% | 17.67 | 10.24 | 7.27 |
| 60% | 150.00% | 23.36 | 13.54 | 9.61 |
The curve steepens. Going from a 10 to a 20 per cent loss more than doubles the gain needed; going from 40 to 50 per cent raises it by half, from 66.67 to 100 per cent. This is the arithmetic case for a diversified fund over a handful of shares: a single company can fall 60 per cent and need 150 per cent to come back, while a broad index has historically fallen less far, because its holdings do not all fall together.
The table describes a portfolio that is left alone. A saver who is still adding money recovers sooner, because each contribution buys at a depressed price.
Keep paying 500 a month into the same fund while its price climbs back at 7 per cent a year (0.5654 per cent a month). By month 64, when the price is back above its old level, 32,000 has gone in and is worth 38,644: a gain of 6,644, or 20.8 per cent, on money that was invested only during the recovery. The pot itself, counting the new money, is back above 100,000 after 31 months, about half the time the price takes, though part of that is your own contributions.
This is why a fall early in a saving life is less damaging than it feels, and why the comparison of regular investing and a lump sum is about expected return rather than about protection.
Now the investor who sells after the fall, holds cash at 2 per cent for two years, then buys back once things feel calmer.
| After two years | Stayed invested | Sold and waited |
|---|---|---|
| Value | 80,143 | 72,828 |
| Units held (100,000 before the fall) | 100,000 | 90,873 |
| Twenty years later at 7% | 310,128 | 281,821 |
The seller locks in the 30 per cent and then buys back at a price 0.8014 of the old one, ending with 9.1 per cent fewer units for good. A 7,315 gap at the point of buying back becomes 28,307 twenty years on. Nothing in the market caused that part of the loss.
Thinking in symmetric percentages: assuming that a 30 per cent rise repairs a 30 per cent fall, or that an average of minus 30 and plus 30 is zero. Returns compound, so they must be multiplied, not added. The same error makes an average return look higher than what an investor actually earned, as worked in CAGR versus the average return.
All figures are illustrative and none is a forecast. This is education, not advice; investing carries the risk of loss.
The same saver's lost decade and sale after a fall are taken to a number in the free workbook and cases for this book.
A 50 per cent loss needs a 100 per cent gain. At a steady illustrative return of 7 per cent a year that takes 10.24 years, at 10 per cent 7.27 years and at 4 per cent 17.67 years. The formula is ln(1 / (1 - loss)) divided by ln(1 + return); actual recoveries are rarely steady.
Because each percentage is measured on a different base. A 30 per cent fall takes 100,000 to 70,000; a 30 per cent rise on 70,000 adds only 21,000, ending at 91,000, still 9 per cent down. Getting back to 100,000 needs 42.86 per cent.
On the illustrative case it is costly. Selling 70,000 after a 30 per cent fall, holding cash at 2 per cent for two years and buying back at 0.8014 of the old price leaves 90,873 units instead of 100,000. The 7,315 gap grows to 28,307 over the next twenty years at 7 per cent.
This article is one calculation from Stock Market Investing for Beginners. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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