Volatility Drag and Loss Recovery
Compare arithmetic and compound annual returns for an alternating pair, or calculate the gain and assumed whole years needed after a loss.
How the two comparisons work
Loss recovery begins at one original unit, applies the entered loss, and asks which percentage gain would restore that original unit. It also counts the first whole year that reaches the original value under a constant assumed annual return, without deposits.
Sequence mode repeats two annual percentage returns in order for ten years. The arithmetic mean adds the two returns and divides by two. The compound annual rate takes the square root of their growth factor product and subtracts one. Their difference is percentage points, not an investment fee.
The displayed paths both start at one unit. One earns the arithmetic mean every year; the other alternates the two entered returns. The constant path is a mathematical comparison, not an available investment. Even year comparisons include complete pairs. An odd year alternating value can exceed the constant path.
No market data, portfolio volatility estimate or forecast is used. The model excludes cash flows, taxes, fees and inflation. Reordering the same two returns leaves a completed pair's terminal product unchanged without cash flows, but changes intermediate balances.
Step by step
- Choose loss recovery or alternating annual returns and enter the corresponding percentages.
- Read the required recovery gain or compare the arithmetic mean with the compound annual rate.
- Inspect whole year recovery assumptions or the normalized ten year table, keeping total returns distinct from annual rates.
Worked examples
A 50% loss leaves 0.5 of the original value. A 100% gain on that remainder restores 1. At an illustrative 7% effective annual return, the balance is about 0.983576 after ten years and 1.052426 after eleven, so the whole year recovery count is 11.
Annual returns of positive 50% then negative 50% give 1 × 1.5 × 0.5 = 0.75 after two years: a total loss of 25%. Their arithmetic mean is 0%, while the compound annual rate is approximately negative 13.397460%, a difference of 13.397460 percentage points. Repeating this pair five times gives 0.2373046875 after ten years; the constant 0% path remains 1.
When to use this calculator
Check how the remaining balance changes recovery arithmetic, and distinguish a simple average from the annual rate implied by compounding two returns.
Use the normalized paths to inspect the exact assumptions. Assess real portfolios, future returns and cash flow needs separately.
Common interpretation errors
Do not confuse a two year total return with a compound annual rate, or percentage point differences with charges deducted from a portfolio.
Do not assume the alternating path is lower in every individual year, that the constant mean path is investable, or that a recovery assumption predicts when a market will recover.
Recovery and compound return formulas
Let d be the loss and a the assumed annual return as decimal fractions.
Required gain = d / (1 − d), for 0 ≤ d < 1
Recovery years = ceil(log(1 / (1 − d)) / log(1 + a)), for 0 < d < 1 and a > 0
Let r1 and r2 be the two annual returns as decimal fractions.
Arithmetic mean = (r1 + r2) / 2
Compound annual rate = sqrt((1 + r1) × (1 + r2)) − 1
Difference in percentage points = 100 × (arithmetic mean − compound annual rate)
Multiply decimal return rates by 100 for percentages. A zero loss requires zero gain and zero years. A 100% loss has no finite recovery gain, and a positive loss does not recover at nonpositive annual growth. Sequence returns may equal but not fall below negative 100%. The visible path uses ten whole years; the calculation API supports zero through one thousand whole years when numeric results remain representable.
Scope and limitations
Returns are hypothetical annual inputs with no market data or forecast. Loss recovery assumes no new cash and an unchanged effective annual return. Sequence mode repeats exactly two returns, starts at one unit and has no deposits, withdrawals, costs, taxes or inflation. Below negative 100% is outside the unlevered growth model. A full loss remains zero. The gap between means is not a fee, probability or measured risk statistic. Numeric overflow, underflow and unsupported whole year counts are explicitly withheld.
Key terms
- Recovery gain
- The percentage increase on a remaining balance needed to restore its original value after a loss below 100%.
- Compound annual rate
- The constant yearly rate with the same ending growth factor as the two entered annual returns together.
- Normalized growth path
- A sequence of hypothetical balances starting at 1, without an assigned currency or new cash flows.
Related comparisons
Maximum Drawdown identifies an observed peak to trough decline in a supplied series. This tool starts with a loss assumption or a repeated pair and does not identify a historical drawdown.
Sequence of Returns can examine withdrawals. CAGR summarizes a complete growth period. Here the two year compound rate and cash flow free paths answer narrower questions.
FAQ
Is a two year loss of 25% the compound annual return?
No. The total return compares the final balance with the initial balance. The equivalent annual compound rate takes the square root of the two year growth factor and subtracts one. A factor of 0.75 means a total loss of 25%, but an annual compound rate of approximately negative 13.40%.
Why does losing 50% require gaining 100%?
The gain applies to the smaller remaining balance. Starting at 1 and losing half leaves 0.5. Adding 50% of 0.5 reaches only 0.75, while doubling the remaining 0.5 restores 1.
Why is recovery time rounded up to a whole year?
The output counts completed annual compounding steps, not an estimated trading date or fractional year. The first whole year meeting the original value is shown. The assumed rate must stay constant and excludes contributions and withdrawals.
What happens at a complete loss or zero assumed growth?
A complete loss leaves no balance to grow, so percentage returns alone cannot restore it. A positive loss also does not recover at zero or negative annual growth. A zero initial loss needs no recovery, regardless of the growth assumption.
Does the order of returns always change the ending value?
Without cash flows, the product of the same two annual growth factors is unchanged when their order reverses. Intermediate and odd year balances can differ. Deposits and withdrawals can make return order affect terminal wealth, but this model has none.
Related tools
Compare observed drawdowns, summarize growth with CAGR, or examine withdrawals separately with a sequence of returns model.