4. Hidden correlation: ten positions are really one

Risk 2: Position sizing

This is the most expensive mistake in the business, and the hardest to see from the inside, because whoever makes it has the opposite impression: he believes he is diversified. He has ten different names, on different screens, bought at different moments. And he has a single bet.

Ten positions, one bet

An investor builds a portfolio out of ten stocks: two semiconductor makers, two enterprise software companies, two digital payments firms, an e-commerce platform, a biotech with no approved product and two technology companies that have grown recently.

On paper they are ten different businesses across five sectors. In risk terms they are almost the same thing: they all make their money far off in the future, they all sink when the discount rate rises, they are all on the same side of the same exposure. Diversifying does not consist of holding different names but of holding different sensitivities to the things that move prices: rates, growth, inflation, credit, currency, liquidity.

The same goes for portfolios that look prudent. A fund holding high-yield corporate bonds, mid-sized bank shares, listed property and emerging market debt has four labels and a single bet: that credit conditions will not tighten.

The arithmetic of average correlation

Here is a calculation worth internalizing. Take ten positions with the same individual swing and the same average correlation between all of them. The portfolio's swing, compared with that of a single position, comes out as follows:

• Correlation zero: the portfolio swings 32% of what one position swings. Diversification is working at full power.
• Correlation 0.3: the portfolio swings 61%. You have already lost half the benefit.
• Correlation 0.6: the portfolio swings 80%. Ten positions have cut your risk by a fifth.
• Correlation 0.8: the portfolio swings 91%. The diversification is practically decorative.

And there is a detail that finishes the idea off. With an average correlation of 0.6, going from ten positions to a hundred takes the swing from 80% down to 77.5% of a single position. Ninety extra names, ninety analysis decisions, ninety lots of commission, to gain two and a half points. When positions share an exposure, adding more does not diversify: it only dilutes your attention.

The effective number of independent bets

Out of that comes a concept that is very useful for talking honestly about a portfolio: the effective number of bets, which answers how many genuinely independent positions the ones you hold are worth.

With the figures above, those ten positions with an average correlation of 0.6 behave like roughly 1.6 independent bets. With a correlation of 0.3, like 2.7. And if in a crisis the average correlation rises to 0.8, the same ten positions behave like 1.2: you have gone from believing you had ten ideas to having, in practice, one.

That number is the honest figure to put alongside the count of positions. Ten names are not ten ideas.

The correlation of calm is not the correlation of crisis

Here is the uncomfortable part. Correlation is not a stable property of two assets: it rises when the market falls. It is a documented and persistent fact in practically every asset class.

The reason is that in calm times each price responds to its own idiosyncratic factors, whereas in a crisis they all respond to the same one: to the forced seller who needs liquidity and sells what he can, not what he wants. When somebody has to close positions at any price, the correlation between the things he is selling approaches one by construction, regardless of fundamentals.

The practical consequence is devastating. A portfolio sized with the correlation matrix of the last two quiet years is calibrated for the scenario in which you do not need it, and miscalibrated for exactly the scenario in which you do. All the diversification that shows up in the spreadsheet evaporates on the day it was going to be of some use.

How to spot it before the market tells you

Without writing a line of code, there are three ways to examine a portfolio:

• An inventory of exposures. For each position, note what it is sensitive to beyond its sector: rates, the price of energy, the dollar, the credit cycle, appetite for risk. If the same word turns up in eight out of ten, you already know.
• Scenarios instead of correlations. Ask what each position would do if rates rose two points, if credit shut down, if the dollar shot up. Scenarios capture the joint dependence that an average correlation smooths away.
• The dominant source of variation. Check how much of the portfolio's daily movement is explained by a single common factor. If that share is high, you have one bet, not ten. Marcos López de Prado has insisted that a correlation matrix estimated on little data is unstable as well, and that grouping assets by similarity before allocating risk produces portfolios that hold up better out of sample.

Real case: the quant quake of August 2007

Between 6 and August 9, 2007, dozens of market-neutral equity funds suffered violent, simultaneous losses. These were funds that, on paper, had no directional exposure to the stock market and that showed very low historical correlations with one another. Several of them lost more in three days than their models considered possible in years.

The most talked-about case was a large Goldman Sachs quantitative fund, whose losses that week were put at somewhere around 30%. Amir Khandani and Andrew Lo later reconstructed the sequence: one participant with problems elsewhere in his business began liquidating his quantitative portfolio; since every fund in the trade held very similar positions, that selling moved prices against the rest; the losses triggered automatic risk reductions, which generated more selling and more losses.

Nobody had taken on that correlation consciously. It appeared in no risk report, because it did not appear in the historical data: it was a latent correlation, switched on by a forced seller. Curiously, most of those positions recovered over the following weeks. The problem was not being wrong about the idea, it was not being able to wait to be proved right.