What exactly is a drawdown
For each date, identify the previous maximum value, including the current date. Drawdown is current value divided by that maximum minus one. Maximum historical drawdown is the deepest such decline within the selected window. If the window starts after an earlier peak, part of the loss disappears from the calculation: the start date is a material assumption.
State separately whether distributions and external contributions are included. A withdrawal of the owner’s money should not look like an investment loss, and a contribution should not look like price recovery. A portfolio with flows needs a consistent value or return method. A simple account-balance line without explanation can mix market results with owner decisions.
One return set, two paths
In the first fictional path, starting at 100 units, returns are −20%, +25%, −20% and +25%. Values are 80, 100, 80 and 100. Maximum drawdown is 20%.
In the second path, the same returns are reordered: two −20% declines followed by two +25% gains. Values are 80, 64, 80 and 100. Maximum drawdown reaches 36%. Both sequences have the same ending value and the same dispersion of periodic returns, but the depth of the temporary loss differs. This is arithmetic, not a forecast of real recovery.
Depth does not replace duration
Two paths with the same maximum drawdown can differ in time spent below the prior peak. A brief fall and a multi-year lack of recovery have different consequences for payments. Preserve peak date, trough date and recovery date when recovery occurs. If capital has not recovered by the end of the window, do not truncate duration as though the event were finished.
Recovery of one surviving asset is not guaranteed. A chart of a surviving company shows one history, not an obligation for the market to restore every prior price. The SEC separately notes that past performance does not predict the future.[27](#source-market-performance) Historical maximum drawdown is not an upper bound on potential loss.
Why calm prices do not prove safety
Sparse trades, model valuations and limited exit capacity can create low observed dispersion. Selling the required size may occur at a very different price. Low measured volatility therefore does not exclude a deep realized loss or delayed access. Before comparing metrics, check how often and on what basis value is updated.
Do not automatically choose the portfolio with the smallest historical drawdown. It depends on period, composition, costs and rebalancing rules. A favorable window can hide the most dangerous event, and a current composition reconstructed with hindsight can remove failed assets. Use a reproducible method, not one promotional number.
A final risk table may include volatility, drawdown depth and duration, liquidity conditions and stress losses. These measures complement one another; they do not add up to a universal score. This material does not set a personally acceptable drawdown or promise a future path inside the historical range.
Clarify whose experience the chart describes
An instrument’s result and a particular owner’s result can differ because of purchase dates, contributions and withdrawals. An index peak before ownership is not a peak experienced by the owner, although it helps describe the market. Conversely, an account balance after a large contribution does not show that an earlier investment loss recovered.
For method comparison, set one starting point and treat flows symmetrically. If one chart includes reinvested distributions and another shows price only, part of the drawdown difference is methodological. Align the data before discussing economic causes.
Preserve measurement frequency. Month-end observations may miss an intramonth fall. That does not make monthly analysis wrong, but it limits the claim. Call it maximum drawdown at the available frequency and window, not a recovery of every real-market movement.
On the chart horizon: what grows as the square root of time
The statement “risk grows more slowly than return” requires defined axes. The standard deviation of a sum of independent identically distributed changes, its ratio to the expected sum, the probability of a positive outcome and a monetary loss are different measures. The variance formula includes covariances; independence cannot be silently transferred to actual returns.[1](#source-cv-covariance)
Build a model: annual log returns Xt = ln(Vt/Vt−1) are independent and normally distributed with mean μ = 0.04 and standard deviation σ = 0.20. There are no contributions, withdrawals, costs or parameter changes. Over T years, the sum has mean μT and standard deviation σ√T. This is a sum of logarithms, not an ordinary average annual return and not rouble risk.
| Horizon T, years | Mean log-return sum | Its standard deviation | Deviation / mean sum | Probability VT > V0 | |---|---|---|---|---| | 1 | 0.04 | 0.20 | 5.00 | 57.93% | | 4 | 0.16 | 0.40 | 2.50 | 65.54% | | 9 | 0.36 | 0.60 | 1.67 | 72.57% |
The mean grows linearly and deviation as the square root of time, but deviation still rises from 0.20 to 0.60. The relative measure σ/(μ√T) declines and is defined for μ > 0. Profit probability is Φ(μ√T/σ), where Φ is the standard-normal CDF. This says nothing about interim drawdown, absolute monetary loss or meeting an obligation before the final date. A logarithmic value axis is also not a log return; an axis is a display choice.
Why probability of profit is not utility
For illustration, assign +1 to any positive final change and −2 to any negative change; in a continuous model the exact-zero probability is zero. Expected score is p − 2(1 − p) = 3p − 2. The table rows give approximately −0.2622, −0.0337 and +0.1772. The vertical axis is a conditional score, not return or welfare.
The score ignores magnitude: losing one unit and 99 units both receive −2. It is not a complete monetary utility function and does not show that losses are “twice as painful” for any person. The coefficient two is an exercise choice. Inflation, interim withdrawals or another evaluation function changes the result.
If changes are positively correlated, covariance terms can make sum variance grow faster than Tσ². If all changes coincide, standard deviation is Tσ, not σ√T. Default, regime change and loss of access are outside the normal model. Before reading a polished graph, write down the metric, units, scale, horizon and assumptions.