The Arithmetic of DrawdownsStart here
Losses and gains are not symmetric, and the asymmetry compounds. Why recovery arithmetic dominates portfolio design far more than expected return does.
The asymmetry, stated once
$$ \text{gain needed} = \frac{d}{1-d} $$
| Fall | Gain needed to recover | Years at 7% p.a. |
|---|---|---|
| 10% | 11.1% | 1.6 |
| 20% | 25.0% | 3.3 |
| 30% | 42.9% | 5.3 |
| 50% | 100.0% | 10.2 |
| 70% | 233.3% | 17.8 |
| 90% | 900.0% | 34.0 |
- The function is convex: it accelerates. Doubling the loss more than doubles the required recovery, and beyond about 60% it stops being a recovery problem and becomes a different portfolio.
- The last column is the one that matters, because it converts an abstract percentage into the thing actually being spent: years. Run your own numbers in the drawdown calculator.
- This is why risk control beats return-seeking in any strategy that must survive. An extra percentage point of expected return is worth far less than not having the 50% year.
Volatility drag: why the average return is not the return you get
$$ g \approx \mu - \frac{\sigma^2}{2} $$
- Arithmetic mean and geometric mean are different numbers, and the gap grows with the square of volatility. +50% then −50% averages zero and leaves you down 25%.
- At 15% volatility the drag is about 1.1% a year; at 40% it is about 8%. This is not a fee and not a cost — it is arithmetic, and it applies to every volatile holding.
- It is the whole story behind daily-reset leveraged products. The drag scales with the square of leverage: at 3× it is nine times larger. The leverage decay calculator shows what a flat-but-choppy year does to one.
- The practical implication: reducing volatility raises compound return even with no improvement in average return at all. That is the entire mathematical case for diversification, and it needs no forecast to work — see diversification.
Sequence risk: the same returns in a different order
- While you are only accumulating, order does not matter. Multiplication commutes; the same set of annual returns produces the same ending value in any sequence.
- The moment money moves in or out, order matters enormously. A bad decade at the start of retirement and the same bad decade at the end produce completely different outcomes from an identical set of returns.
- Why: a withdrawal during a drawdown sells more units to raise the same cash, and those units are permanently gone from the recovery.
- The mirror image works in your favour when contributing: regular buying into a falling market acquires more units, which is the honest version of the case for spreading purchases — see the lump-sum versus phased calculator, which also shows that on average the lump sum wins.
- The withdrawal calculator is deterministic on purpose: it shows the arithmetic clearly, and its final note says exactly what it leaves out.
Why the tail is fatter than the model says
- Normal-distribution risk numbers understate. A parametric VaR quotes the loss exceeded on one day in twenty; historically, the days beyond it are both more frequent and much larger than the model implies.
- Losses cluster. Volatility is autocorrelated — bad days arrive together, which is why a "one in a hundred years" event can appear three times in a fortnight without the model being wrong about any individual day.
- Expected shortfall answers the better question: not "how bad is the threshold" but "how bad is it when the threshold is breached". Both are in the VaR and expected-shortfall calculator, with its assumption stated on the page.
- The honest reading of any tail number is as a floor on how bad things can get, never as a bound.
What the arithmetic implies for construction
- Size so that the worst plausible loss is survivable, then check what expected return that leaves — not the other way round. The position-sizing playbook is this inversion in full.
- Prefer reducing the tail to raising the mean. The convexity of the recovery function means avoided losses are worth more than equivalent gains.
- Rebalancing is a volatility-reduction tool before it is anything else, and its benefit shows up in the geometric return rather than the arithmetic one.
- Leverage interacts with all of this multiplicatively. It scales the mean linearly and the drag quadratically, which is why the optimal leverage for compound growth is finite and usually much lower than intuition — the point Kelly makes formally.
Information and education only. Every figure on this page is arithmetic on illustrative inputs, not a forecast and not a claim about any market's history or future. Nothing here is advice or a recommendation about how much risk anyone should take.