The math of loss and recovery: a halving needs a doubling

It is widely known that a −50% loss takes +100% to make up. Less well known is that the same asymmetry works on every daily move, so the bigger the swings, the wider the gap between the average return and the return that actually compounded. This article covers the arithmetic of loss and recovery, how much volatility takes out of compounding, and how long SPY, QQQ and TQQQ actually took to recover from their 2000, 2008, 2020 and 2022 highs. It also compares recovery for a lump sum and for monthly investing. Figures are as of September 30, 2026.

The asymmetry of loss and recovery

If a price falls by L, what is left is 1 − L, and the gain needed to get back is 1 ÷ (1 − L) − 1. For small losses the two are close, but as the loss grows the required gain grows fast.

A −10% loss needs +11.1% to make up, −30% needs +42.9%, −50% needs +100% and −80% needs +400%. At 10% a year, making up −50% takes 7.3 years and −80% takes 16.9 years. QQQ fell −83.0% with dividends from March 27, 2000 to October 9, 2002, and even though it then rose 15.4% a year it took another 12.4 years to regain its high.

Volatility drag: arithmetic and geometric means

A +10% year followed by −10% leaves −1%, not 0%. The arithmetic mean of the two years is 0%, but the geometric mean, the rate the money actually grew at, is negative. The gap is roughly the square of volatility (standard deviation) divided by two, and is called volatility drag. Triple the same swings to +30% and −30% and the result is −9%; repeat ten times and ±10% gives −9.6% while ±30% gives −61.1%.

Using annual returns with dividends, SPY (1994 to 2025, 32 years) had an arithmetic mean of 12.3% and a geometric mean of 10.7%. QQQ (2000 to 2025) had 12.5% against 8.3%, and TQQQ (2011 to 2025) 60.4% against 39.2%. Annual volatility measured from daily returns was 18.7% for SPY, 26.9% for QQQ and 61.1% for TQQQ. The higher the volatility, the wider the gap between the two means. If an "average return of X%" is an arithmetic mean, the account's actual return is lower.

Leveraged ETFs are 3x for one day

TQQQ targets three times the Nasdaq-100's daily return, and the issuer's prospectus states that over periods longer than a day the return can differ significantly from three times. TQQQ launched on February 9, 2010; from February 11, 2010, where this site's data begins, to September 30, 2026, QQQ returned 19.6% a year. Three times that is 58.7%, but chaining three times QQQ's daily return gave 50.2% a year even with no costs, and actual TQQQ returned 42.9%. The first gap is volatility drag; the second is fees, financing costs and tracking error.

The gap looks bigger in a sideways market. From November 19, 2021 to December 13, 2023, QQQ with dividends was flat at +1.2%, but on the same day TQQQ was -44.0% below its November 2021 high. Chaining three times QQQ's daily return, with no costs, gave -32.1%. The underlying index can regain its high while the leveraged ETF does not.

Actual recovery times (as of September 2026)

The high in the table is the prior record close with dividends included; for QQQ in 2007 to 2008 only, it is the 2007 high, not the 2000 high. Price recovery is the first day the split-adjusted close passed the high again, and recovery with dividends is the same for prices with dividends reinvested. QQQ's 2022 price high was on November 19, 2021, so the dates differ. The last column assumes $500 went in on the first trading day of every month from the first trading day of the high's month, and gives the first day after the index bottom when the balance was at or above the money put in.

Dividends brought recovery forward. SPY regained its 2000 high in 7.2 years on price and 6.6 years with dividends; for QQQ it was 16.4 and 14.9 years. TQQQ, which pays little, showed almost no difference. TQQQ's 2022 decline of −81.7% was 2.3 times QQQ's (−35.1%), and its recovery took 3.0 years.

Why monthly investing recovered sooner

Putting in the same amount every month buys more shares when prices are low. The average purchase price ends up well below the high, and the balance passes the money put in long before the index regains its high. Monthly investing in QQQ from March 2000 passed the money put in on January 7, 2004, while a lump sum put in the same month passed it only on August 25, 2014.

The price is the size of the loss in dollars. With monthly investing the money put in keeps growing, so a later decline hits a larger sum. Monthly investing in QQQ from March 2000 had $4,324 against $9,500 put in on September 21, 2001, a −54.5% loss; after passing the money put in, it went under again in the 2008 decline, and the last day below was Jul 17, 2009. Monthly investing in SPY from March 2000 had $34,433 against $54,500 on March 9, 2009. At −36.8% it was deeper than at the 2002 bottom (−33.6%), and the $20,067 shortfall was 3.7 times the $5,369 of the 2002 bottom. The last day below the money put in was Jul 6, 2010.

What research and filings say

• Loss aversion: in a 1992 study, Kahneman and Tversky estimated that a loss feels about 2.25 times as large as a gain of the same size (Investor psychology). The +100% needed after −50% is double in numbers, and feels farther still.
• Leveraged ETF warning: in a joint investor alert on August 18, 2009, the US Securities and Exchange Commission (SEC) and FINRA wrote that leveraged and inverse ETFs, which reset daily, can produce results over longer periods that differ significantly from their target multiple because of compounding.
• Recovery is not a law: Japan's Nikkei 225 first closed above its December 29, 1989 close on February 22, 2024 (about 34 years on price, CNN).

When the idea holds and when it fails

• The asymmetry always holds: it is arithmetic, so it applies to any stock or market. What changes is the gain and the time available for recovery.
• Recovery time was not set by the depth of the fall alone: SPY regained −33.7% in 5.7 months in 2020, and −47.5% in 6.6 years after 2000 to 2002. How fast prices rose after the fall made the difference.
• The fast recovery of monthly investing only holds if the investing continues: stopping during a decline removes the shares bought near the bottom (QQQ monthly investing backtest).
• A leveraged ETF can take longer to recover than its index: TQQQ regained its 2021 high only on December 4, 2024, 11.7 months after QQQ regained its own on December 13, 2023 (with dividends, TQQQ vs. QQQ and QLD monthly).

Related concepts and search terms

• Volatility drag, volatility decay: swings push the geometric mean below the arithmetic mean. For leveraged ETFs it is also called negative compounding or decay.
• Recovery time, time to breakeven: the time from a high back above that high. Counting from the start of the fall, it is also called the underwater period.
• Maximum drawdown (MDD): the largest fall from a high to a low.
• Sequence of returns risk: with the same average return, the outcome for someone adding or withdrawing money depends on when the big losses arrive. The larger dollar loss in the later decline in the monthly cases above is one example.

Search terms: volatility drag, volatility decay, leveraged ETF decay, recovery time after crash, arithmetic vs geometric mean, gain needed to recover a loss, how long to recover from a 50% loss, Nasdaq recovery time, S&P 500 recovery time.

Limits of these numbers

• These are periods in which broad US index ETFs all recovered. They cannot be carried over to other markets or single stocks.
• TQQQ launched in 2010 and did not go through the 2000 and 2008 declines. How it would have fared in them is not calculated here.
• Taxes, exchange rates and returns in won are left out. Monthly investing used $500 a month and a 0.15% cost per trade.
• The volatility drag formula is an approximation. The actual gap depends on the shape of the return distribution and the period.