The Arithmetic of Drawdowns

Start with a fact most investors know and few have internalised: a loss and the gain required to undo it are not the same size. They are not close to the same size once the loss becomes serious, and the relationship gets worse in a way that intuition does not anticipate.

Read the last column rather than the second. A 50% fall does not cost you 50%. It costs you roughly six years — six years of compounding spent returning to a position you already occupied. For an investor with a thirty-year horizon that is a fifth of the entire runway, surrendered in a single episode.

A related and more insidious consequence: the arithmetic average of a series of annual returns is systematically higher than the compounded return actually realised, and the gap grows with volatility.

The canonical demonstration is two years: up 50%, then down 50%. The arithmetic average is zero. The realised outcome is −25%. Nothing has been mismeasured; the two numbers simply answer different questions. The average answers 'what happened in a typical year'. The compounded figure answers 'what do I have', which is the only one that buys anything.

The practical consequence is that reducing volatility raises compounded return even with no improvement in average return at all. This is the closest thing to a free lunch in portfolio construction, and it is the mathematical justification for diversification, position limits and rebalancing — none of which require anyone to forecast anything.

Everything above assumes a single lump sum left alone. Almost nobody invests that way. Once money is going in or coming out, a third asymmetry appears — and it is the one that decides retirement outcomes.

Take two investors with identical portfolios earning an identical set of annual returns, in opposite orders. Left alone, they finish in exactly the same place: multiplication is commutative. But an investor contributing monthly does much better when the bad years come first, because contributions buy more units at lower prices. And an investor withdrawing monthly does much worse when the bad years come first, because units are being sold at depressed prices and are permanently unavailable for the recovery.

1. 1

   In the accumulation phase, early weakness is a benefit

   Counter-intuitively, a long flat or falling market early in an investor's contributing life is among the best things that can happen to them, provided the contributions continue. The most damaging response to it is the instinctive one: stopping.

2. 2

   In the withdrawal phase, early weakness is the central risk

   A bad first five years of retirement can be unrecoverable regardless of what the subsequent twenty years do, because the capital that would have participated in the recovery was already spent.

3. 3

   The bridge between them is held liquidity

   A reserve covering two to three years of planned withdrawals converts a forced sale at the bottom into a choice. That reserve is not an under-invested drag — it is the instrument that makes the rest of the portfolio survivable.

4. 4

   Risk should fall as the withdrawal date approaches, then can rise again

   Exposure to sequence risk peaks around the transition, not across retirement as a whole. A portfolio ten years into a withdrawal phase has already survived the window that mattered most.

All of which suggests that the risk statistics most commonly reported are not the ones most worth reading. Annualised volatility is a good comparative measure and a poor experiential one: nobody has ever experienced a standard deviation. What people experience is a peak-to-trough fall, in currency, over a specific number of days.

Prospera's own Portfolio Intelligence methodology takes this position explicitly, and it is worth quoting the reasoning rather than the numbers: the maximum drawdown row is described as the most useful row in the document, and it is deliberately restated in rupees on a stated starting balance rather than left as a percentage. The reason is that a percentage is an abstraction and a rupee figure is a decision — it is the number that tells you whether you would actually have held on.

The arithmetic of drawdowns does not argue for timidity. A portfolio held too conservatively fails in a slower and less visible way, and inflation is a drawdown that never shows up on a chart.

What it argues for is that risk should be chosen at a level you can hold through, and that the level is defined by the worst realistic fall rather than the expected return. Return assumptions are estimates about a future nobody can see. The recovery arithmetic is not an estimate — it is fixed, it is knowable in advance, and it is the part of the problem that will behave exactly as predicted.

Choose the drawdown you can survive. The returns will be whatever they are going to be, but the survival is a decision you get to make now.