1. The asymmetric arithmetic of recovering from a fall

Risk 3: Drawdown and ruin

Losing money and getting it back are not symmetrical operations. The sentence sounds obvious and almost nobody does the arithmetic. If your portfolio falls 20%, intuition says you need 20% to get back to where you started. You need 25%. That five-point gap is not an accounting detail: it is the law that governs everything that comes after it in this level.

The calculation intuition gets wrong

You have 250,000 euros and you lose 20%. You are left with 200,000. To get back to 250,000 you have to make 50,000, and those 50,000 on the 200,000 you have left are 25%. The reason is that the loss is calculated on the capital you had and the recovery on the capital that survived. They are two different denominators, and the second is always the smaller one.

The general rule fits on a single line: the gain required is the loss divided by what is left. A 20% fall leaves 0.80 and demands 0.20 divided by 0.80, that is 25%. A 50% fall leaves 0.50 and demands 0.50 divided by 0.50: 100%.

The table, walked all the way through

These figures are worth carrying in your head, because they change the way you look at any position:

A fall of 10% demands a rise of 11.1%. Next to nothing.
Of 20%, 25%. Annoying, but that is a normal year in equities.
Of 30%, 42.9%. Now we are into extraordinary-year territory.
Of 40%, 66.7%.
Of 50%, 100%: the capital has to double.
Of 60%, 150%.
Of 70%, 233%.
Of 80%, 400%.
Of 90%, 900%: you have to multiply by ten.

Notice that the first half of the table is boring and the second half is terrifying. Between a 10% and a 30% fall, the requirement rises by thirty-two points. Between 70% and 90%, it rises by almost six hundred and seventy.

Why the asymmetry accelerates

The divisor in the formula is what is left, and that divisor shrinks as the loss deepens. While you still hold almost everything, dividing by 0.9 barely changes a thing. When only a tenth is left, dividing by 0.1 multiplies by ten. In the limit, with capital tending to zero, the rise required tends to infinity: that is why there is no possible recovery from zero, however good the strategy that comes next.

You will see it better point by point. The first percentage point you lose costs you 1.01 points of recovery. The point that runs from an 89% loss to a 90% loss costs you 91 points: the requirement jumps from 809% to 900%. The same point of loss is ninety times more expensive at the end of the journey than at the start. You are not on a linear scale, you are on a hyperbola.

Out of that comes the practical conclusion of the whole level: avoiding a large loss is worth far more than capturing a large gain. It is not prudence of temperament, it is arithmetic. Making 900% is a rare event that happens to almost nobody; losing 90% is a frequent event in concentrated, leveraged portfolios. And one does not cancel the other out, because one of the two is easy and the other is not.

Arithmetic mean and geometric mean

Here is the formal translation of the asymmetry, and it is the reason so many brochures are technically true and practically misleading.

A strategy gains 50% one year and loses 50% the next. The arithmetic mean of those two returns is zero. What you actually collect is something else: 1.50 multiplied by 0.50 gives 0.75. You have lost 25% over two years, 13.4% a year compounded. Capital does not add returns up, it multiplies them, and that is why what governs your wealth is the geometric mean, always smaller than or equal to the arithmetic one.

The distance between the two depends on volatility, and grows with its square. As a practical approximation, the geometric mean falls below the arithmetic one by roughly half the variance: with a volatility of 15% a year the toll is around one percentage point a year; with a volatility of 40% it comes close to eight points. The same average return, delivered with more swing, hands over far less money.

Two investors, the same decade

Elena makes 12% every year for ten years. Marcos makes 20% for nine years and loses 50% in the tenth. Marcos's arithmetic mean is 13%, a point better than Elena's, and his record looks far more impressive in a presentation.

Elena ends up with 3.11 times her starting capital. Marcos ends up with 2.58 times. The single bad year ate nine excellent years and there was still some to spare. If instead of 50% he had lost 70%, he would end up with 1.55 times his capital, below someone who had earned a modest 5% a year with no bumps.

Real case: fifteen years under water

The Nasdaq Composite hit its bubble peak in March 2000, at around 5,048 points, and bottomed out in October 2002 near 1,114: a fall of roughly 78%. Apply the formula. Recovering that 78% did not demand 78%, it demanded around 353%. The index did not return to that nominal level until the spring of 2015, fifteen years after the peak, and in real terms, once inflation is stripped out, considerably later.

Compare it with the financial crisis. The S&P 500 fell close to 57% between October 2007 and March 2009. The rise required was 133%, and it arrived in a little over four years: the index set a new high in 2013. A 78% fall and a 57% fall do not differ by twenty-one points of pain, they differ by some ten years of waiting under water.

None of this is a recommendation about what to buy. It is the yardstick you will use to measure every strategy you come across from now on: not by what it promises to earn, but by what it would demand to recover if the road turns.