Drawdown and Volatility Drag

A 50% fall needs a 100% rise to undo. The same asymmetry is running every year, with no crash required.

Worked example — Drawdown and Volatility Drag

Arithmetic mean	11.86%
Compound annual growth	10.02%
Drag (mean − compound)	1.84%
Half the variance	1.86%
Worst drawdown	64.8%
Gain needed to recover	+183.8%
Volatility drag	1.84%

These are the values the calculator shows when the page opens. Change any input above to recompute them.

Loading interactive simulation...

The drag is not a crash — it is the price of the ride 🖖

Two portfolios can earn the same average return and end at very different places, with no disaster in either. The gap is close to half the variance, and it is why a smoother 8% beats a wilder 8%. Measured on this data the approximation is tight: the ratio of the actual drag to half the variance stays between 0.90 and 1.09 over 505 combinations of equity share and window — every share from 0 to 100%, across the full record and four sub-periods. It also explains something that looks like a paradox — the arithmetic mean of a series is the number you would quote, and the compound rate is the number you would receive, and only the second one buys anything.

The shallowest portfolio is not the one with no equities 🖖

Set the equity share to 0 and the deepest fall reads 21.5%, bottoming in 2022. Raise it and the number goes down — 18.8% at 10%, then 17.9% from 14% all the way to 28% — before it starts climbing: 18.7% at 30%, 25.6% at 40%, 64.8% at 100%. The trough year flips between 28 and 29, from 2022 to 1931. Two different disasters are competing for the title, and the shallowest mix on this record is the one holding enough of each asset that neither wins. The two ends of the equity field do not show that; only the 1% steps between them do.

Thirteen years in which the drag cost more than the market paid 🖖

Click Lost decade, 2000 at 100% equities and two of the four cards disagree in a way that is easy to read past: the compound rate is 1.63% a year and the drag is 1.78%. Volatility took more than the portfolio delivered. Nothing prevents that — the drag is about half the variance and the compound rate is whatever the mean has left after paying it, so any window whose variance is large against its mean does the same. Cut the equity share to 60% over those same thirteen years and the drag falls to 0.36% against a compound 4.57%: from more than the whole return to about 8% of it.

Problem solved in full

Four annual returns that fix the depth of the 1929–1932 crash 6 steps
Four annual returns fix the depth of the 1929–1932 crash. The gain that undoes it is a different number, and the time it should take is a third one — derive all three. This is the tool's opening state: 100% equities across the whole 98-year record from 1928, where the Compound annual growth card reads 10.02%.
1. A year multiplies your money by 1 + r and does nothing else to it, so four bad years in a row are a product and not a sum. Convert the returns to factors before anything else. They are the S&P 500 annual total returns for those years, from the dataset named under Where the numbers come from.
  
  $r_{1929 \dots 1932} = - 8.30%, - 25.12%, - 43.84%, - 8.64% ⟹ 1 + r = 0.9170, 0.7488, 0.5616, 0.9136$
2. Multiply. Then notice what is missing: 1928 returned +43.81% and it is deliberately absent, because a drawdown is a ratio to the high-water mark and whatever that mark was divides out. How high you climbed first can make the hole neither shallower nor deeper. The Worst drawdown card agrees, and its hint names the same two years.
  
  $\frac{V _{1932}}{V _{1928}} = 0.9170 \times 0.7488 \times 0.5616 \times 0.9136 = 0.352305 ⟹ L = 1 - 0.352305 = 64.8%$
3. Climbing out means multiplying back up to 1, so the gain required is the reciprocal minus one. Take it from 0.352305 and not from the printed 64.8%: one decimal place is enough to display a depth and not enough to invert one. The rounded version gives 184.1%, a third of a point above what the card says.
  
  $g = \frac{1}{0.352305} - 1 = 2.83845 - 1 = + 183.8% (\frac{1}{1 - 0.648} - 1 = 184.1%)$
4. That is the asymmetry in general. Losing half needs a double; losing 64.77% needs almost a treble. Every gain is measured against the smaller number, so as the loss grows the denominator shrinks and the requirement runs away from it.
  
  $g (L) = \frac{1}{1 - L} - 1 : L = 0.5000 \Rightarrow + 100.0%, L = 0.6477 \Rightarrow + 183.8%$
5. Now put a clock on it. The compound annual growth rate is the single rate that reproduces the whole 98-year window, so ask how many years at that rate multiply by 2.83845.
  
  $n = \frac{ln 2.83845}{ln 1.1002} = \frac{1.04326}{0.09549} = 10.9 yr$
6. Open How long it took to get back and read the year. The 1928 high returns in 1936 — four years past the 1932 trough, not eleven.
  
  $1936 - 1932 = 4 yr \neq = 10.9 yr$
Answer

64.8% down, +183.8% to get back, and it took four years where 10.02% a year predicts 10.9. A compound rate is fitted to a whole window, which is exactly why it cannot time any part of one. 1933 returned +49.98%, 1935 +46.74% and 1936 +31.94%, with 1934 flat at −1.19%; those four factors multiply to 2.869, clearing the 2.838 required, at an implied 30.15% a year. Recoveries do not arrive at the average rate. They arrive in two or three enormous years with an ordinary decade around them — and the concentration is the fragile part: strike 1933 out and the remaining three years multiply to 1.913, well short of 2.838, so one missing year of that kind pushes the whole recovery past 1936. The tool prints an average rate on a card and a calendar year inside a collapsed panel. It computes neither the 10.9 years nor the distance between them, and that distance is the reason the two are worth reading together.

$L = 1 - k = 1929 \prod 1932 (1 + r_{k}) = 64.8%, g = \frac{1}{1 - L} - 1 = + 183.8%, \frac{ln ( 1 + g )}{ln 1.1002} = 10.9 yr \neq = 4 yr$

Learning path

Risk, measured

Leads to Order matters the compound rate, and why it always sits below the arithmetic mean by roughly half the variance.

References (3)

Drawdown treated as the risk measure itself rather than as a symptom of variance: A. Chekhlov, S. Uryasev and M. Zabarankin, "Drawdown Measure in Portfolio Optimization." International Journal of Theoretical and Applied Finance 08(1), 13–58, 2005.
And a recent refinement of the same measure, for anyone who wants the conditional version: D. Rossello and S. Lo Cascio, "A refined measure of conditional maximum drawdown." Risk Management 23(4), 301–321, 2021.
Where the log-growth argument behind the compound rate comes from — why the geometric mean, not the arithmetic one, is what a repeatedly reinvested stake actually earns: J. L. Kelly Jr., "A New Interpretation of Information Rate." Bell System Technical Journal 35(4), 917–926, 1956.

By the 42Math team · Updated 2026-09-03

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

All equities, 1928 on - The full record at 100% equities, whose deepest fall below the high-water mark is the 1929-1932 collapse.
60/40, 1928 on - The same 98 years with 40% in 10-year Treasuries, which cuts the deepest drawdown substantially.
Bonds only, 1928 on - Treasuries alone, whose worst drawdown in nominal terms is not 1929 or 2008 but 2022 — though in real terms the long inflation of 1946 to 1981 cost far more.
Lost decade, 2000 - 2000 to 2012 at 100% equities: two deep falls in one window and very little compound growth to show for it.