The fine print of trading and investing

Quiz · Risk Management · Beginner

Down 50%, Up 100%: A Loss Recovery Math Quiz

Loss recovery math is short and unforgiving. Each question gives you a loss, a streak or a pair of returns; work out where the money actually ends up.

AI-assisted, reviewed by the The Margin Report human editor: John James → Marked as you go Published

What it tests

Whether you can work out the gain needed to recover a loss, follow a string of returns in dollars and see what leverage and averages do to the result.

A $6,000 account needs $4,000 to get back to $10,000. That is a 66.7% gain to repair a 40% loss. The two numbers look like they should match. They never do, because the loss comes off the bigger balance while the recovery has to be earned on the smaller one, and every extra point of loss shrinks the base the recovery works on, so each one adds more to the required gain than the point before it did.

This matters in money terms. A risk rule that allows a deep drawdown is really a rule that commits you to an outsized gain later.

The one formula

The gain you need equals the loss divided by what is left.

Every question below reduces to that division or to multiplying a string of returns together. Dollars settle it.

How to answer

Turn each percentage into dollars on a round starting balance. Follow the balance step by step, and write each new balance down before you apply the next return to it. A calculator is fine, and so is a pencil. Resist any answer you reach in your head in under a second, since shortcuts with percentages feel reliable and are the fastest way to a number that does not match the account.

Using a wrong answer

Take a miss seriously. The same error in a real account produces risk limits that allow losses much larger than they sound, and nobody notices until the account is already deep in the hole and the arithmetic of the way back has turned against it. Rework the question in the loss recovery calculator. The column a 50% loss needs a 100% gain makes the case for building risk limits around this arithmetic. The page on maximum drawdown shows how to measure the worst fall from a peak in your own account, and the lesson on drawdown limits and getting back to even turns the numbers into a written limit. Come back to the questions after that, with the working hidden.

Pick an answer to mark it

The questions

  1. The situationA $10,000 account falls 20%, to $8,000.

    1What percentage gain brings it back to $10,000?
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    C: 25%. The account needs $2,000 back, and it has to earn that on $8,000. $2,000 / $8,000 = 25%. The recovery is always measured from the smaller balance, so it takes a larger percentage than the loss.

  2. The situationA $10,000 account loses 10% in one month and gains 10% the next.

    2Where does it end up?
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    B: Down 1%, at $9,900. The 10% loss takes it to $9,000. The 10% gain is 10% of $9,000, or $900, which brings it to $9,900. Equal percentages in opposite directions always leave you behind, because the gain works on a smaller base.

  3. The situationAn account that peaked at $40,000 falls 50%, to $20,000.

    3What gain returns it to the peak?
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    A: 100%. Getting from $20,000 back to $40,000 means adding $20,000 to $20,000, which is a 100% gain. Half the money is gone, so what remains has to double. The recovery formula gives the same answer: 0.50 / 0.50 = 1.00.

  4. The situationA trader takes three losses in a row, each 10% of the account's balance at the time.

    4What gain is needed to get back to the starting balance?
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    D: About 37.2%. Each loss leaves 90% of what was there: 0.90 x 0.90 x 0.90 = 0.729, so the account is at 72.9% of its start, down 27.1%. Getting back needs 1 / 0.729, about 1.372, which is a 37.2% gain. 27.1% is the size of the hole; 37.2% is what fills it.

  5. The situationYou put $20,000 of your own money into a margin account and borrow $20,000, buying $40,000 of stock. That is 2:1 leverage. The stock falls 25%.

    5What happens to your equity?
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    B: It falls 50%, to $10,000. A 25% fall takes the stock from $40,000 to $30,000. The $20,000 loan is unchanged, so equity drops to $10,000, half of what you put in. With the loan still in place, the stock then has to rise from $30,000 to $40,000, about 33.3%, just to restore the equity.

  6. The situationA hypothetical fund returns +50% in its first year and -50% in its second. Its report shows an average annual return of 0%. You invested $10,000 at the start.

    6What is the $10,000 worth after two years?
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    C: $7,500. The first year takes $10,000 to $15,000. The second takes half of $15,000, leaving $7,500. The simple average of +50% and -50% is 0%, and the money is down 25%. To see what happened to the account, multiply the returns: 1.5 x 0.5 = 0.75.

Readers also ask

What is the formula for the gain needed to recover a loss?

Divide the loss by what is left. As decimals: gain needed = loss / (1 - loss). A 30% loss leaves 0.70 of the money, so the gain needed is 0.30 / 0.70, about 42.9%. The larger the loss, the faster that number climbs, which is why small losses are far easier to repair.

Why do percentage gains and losses not cancel out?

Each percentage is measured from a different starting amount. A loss is taken from the larger balance and the gain that follows is earned on the smaller one, so the same percentage brings back fewer dollars than went out. Only dollars, or returns multiplied together, show where the account really stands.

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